Spuriae

This is more a side effect of large driver headphones tending to have larger enclosures, which leads to more pinna activation. When a small driver is used in a headphone with the design approach of a large ventilated enclosure you still get a large soundstage. Compare the HD660S, HD700, and HD800: the HD700 and 800 are more similar as far as soundstage, impact, and "bigness" go, than the HD700 and HD660S are. This despite the 660S and 700 using very similar (if not the same) 40mm drivers; the similarity in enclosures between the 700 and 800 has a much larger effect on the headphone's sound. Meanwhile the HD700-based driver of the HD660S sounds like it belongs in the HD650 family when used in the same enclosure.
 
In enclosure-agnostic terms, a large driver tends to have better bass performance but worse treble performance than a smaller one. Given the significant variations in enclosure design in the market as a whole, however, there isn't a strong correlation between audio quality and driver size that would make any comparisons on that basis worthwhile.

Driver size affects how you design a headphone, but has minimal effect on how the headphone will sound in the end. Enclosure design has a much larger impact on sound than the driver does.

For an effective comparison, you would need to find a recording of your headphones and compare them to a recording of the other headphones. A comparison with listening on your headphones directly only works if you listen to the other headphones directly too.
 
Keep in mind this still isn't a perfect comparison - for instance, if your headphones have less treble than usual, they could make a recording of a headphone with too much treble sound unnaturally good - but it's much better than working off a single recording alone.

Are you looking for a headset or just the headphones? Will this be for gaming, on the go, etc.? Does it need to be closed-back, or is open back an option?
 
Not knowing much about your use case, consider the HyperX Clouds for $35, which are probably the best all-around conventional closed-backs at the $35 price point. The Koss offerings that @Skipple mentioned (and the similar KSC75 ear hook from the same family) are probably the best-sounding headphones within your budget, but if you don't have any other headphones I think the quality of life/durability/comfort advantage on the Clouds is worth the tradeoff in sound quality.

Difficult price point for over-ears. Much easier on the IEM side. 
Perhaps the Koss Porta Pro Utility. These are typically $50 but on sale right now for $25. The Koss KPH40 are on sale too. 
 

Since you don't have measuring equipment, you haven't solved some equations and you don't know exactly what you are doing, the result will be completely random.
 
Capacitors are cheap. Just play around with them and use what you like most.

Open backs are roughly as loud on the outside as closed-back headphones taken off your head and rotated to have the cups facing outwards. You can try that with your current headphones to get an idea of whether it's too much sound leakage.
 
FWIW I've had roommates with open backs and it has never been an issue.

I was skeptical, so I tested it. After all, tactile transducers exist; why can't the same effect be obtained in normal headphones, right?
You're absolutely correct. I was wrong. There is no meaningful haptic effect. Even with the drivers full-on physically clipping I felt nothing tactile. While I was able to hear below 20Hz as a distinct tone, it would probably never be audible with other sounds playing at the same time.

Most motherboards actually output the most power at around 80Ω due to a 75Ω output resistor.
 
Impedance as almost no correlation with difficulty to drive in practice, and efficiency/sensitivity is the spec that really matters. In general motherboards can get most headphones loud enough, but the sound quality will be reduced due to the motherboard's high output impedance.
 
 
 

That the speakers can`t do 20Hz and the headphones can, is normal. Through the very small and sealed amount of air between your ears and the driver, it basically has a more or less direct coupling between the two and very low frequencies get less conducted as a wave, but more as a pressure change in the whole volume. Because of that its very easy for headphones to get these low frequencies into your ear.
 
Your loudspeakers would need to move enormeous amounts of air to achieve the same. Thats the reason why subwoofers that actually work for very low frequencies at a high enough volume are huge compared to the ones that work down to around 40Hz like most HiFi loudspeakers.

In the context of headphones, you won't be feeling infrasonic in any meaningful way from a headphone. 

I also am kind of "ehh" on claims about supersonic frequencies. In A LOT of audio recordings sound is rolled off past ~20KHz so there's literally no signal past the roll off point (err negligible). 

A frequency response spec given only as a range between two frequencies is completely meaningless, since you can technically produce any range of frequencies with almost any device – it's the loudness at that frequency that really matters. For instance, Airpods can technically produce bass down to 20Hz, but bass they produce at 20Hz is too quiet to be meaningful. It wouldn't be a lie to spec their lower range to 20Hz, or 5Hz, or whatever arbitrarily high or low number; it just wouldn't tell you anything about how they sound. 
 
Being able to produce frequencies beyond the conventional audible range has some benefits; you can feel lower frequency as haptics and can perceive higher frequencies are improved timing accuracy. Practically speaking, however, it's rare for these qualities to be relevant or even audible in everyday music listening, given how many other factors have a greater effect.

You will, as the adapter has a pretty low output impedance. If you still have a phone with a headphone adapter (rip) or access to one, you could try comparing your PC and the phone to each other. Phones also typically have pretty low output impedance.
 
It always depends on the person and the specific setup if you notice the differences in sound. So it is hard to tell you if it is actually going to be worth it, without having listened to the options first.
 
BTW, if you start going down the audiophile rabbit hole, be wary of people claiming things like "buying this DAC / AMP will make your headphones sound 50% (or insert arbitrary percentage here) better". That is usually a pretty subjective remark, while the sound actually doesn't change a whole lot.

Most motherboards actually output the most power at around 80Ω due to a 75Ω output resistor.
 
Impedance as almost no correlation with difficulty to drive in practice, and efficiency/sensitivity is the spec that really matters. In general motherboards can get most headphones loud enough, but the sound quality will be reduced due to the motherboard's high output impedance.
 
 
 

Most motherboards actually output the most power at around 80Ω due to a 75Ω output resistor.
 
Impedance as almost no correlation with difficulty to drive in practice, and efficiency/sensitivity is the spec that really matters. In general motherboards can get most headphones loud enough, but the sound quality will be reduced due to the motherboard's high output impedance.
 
 
 

How many OHMs were the first pair and how many OHMs is the new pair?

How are you measuring the impedance? Impedance should vary based on frequency.
 
If you're doing a resistance measurement, the capacitor acts like an open circuit with a DC input so you're essentially measuring the resistance of the wire in the woofer. 6Ω (resistance) on an 8Ω (nominal impedance) woofer is normal.

How are you measuring the impedance? Impedance should vary based on frequency.
 
If you're doing a resistance measurement, the capacitor acts like an open circuit with a DC input so you're essentially measuring the resistance of the wire in the woofer. 6Ω (resistance) on an 8Ω (nominal impedance) woofer is normal.

Here's a thread where people talk about software for simulating cross over designs. You should be prepared to spend A LOT of time designing a cross over and probably a decent amount of cash since the first design or two probably will be "ehh"

https://www.audiosciencereview.com/forum/index.php?threads/crossover-software-for-windows-linux-macos.26180/

If you're doing this "right" you'd be designing a custom cross over for the unique combination of woofer+tweater+speaker cabinet that you have. 
You shouldn't just be replacing the capacitors and resistors like for like in this case because you're using a different woofer from what the speaker was designed for. 
When you changed out the busted woofer from your speaker for a different, non-identical woofer (which might have a similar F3 but likely has very different frequency response and sensitivity characteristics) you changed the characteristics of the overall speaker. The old crossover design won't necessarily work anymore so the parts inside the speaker's crossover network (capacitors, resistors) would all ideally be swapped out. 
It's like changing one wheel on a car for a bigger wheel... EVERYTHING else needs to be adjusted (the wheel on the other side needs to be swapped, you might need to redo the brakes, you might need to change the calibration for the speedometer and odometer, etc.)
You shouldn't be building your first crossover from scratch.
Ideally you're following a guide and using a pre-built kit. Or you're replacing a part like for like. 
This is NOT the same as getting a mismatched woofer that happens to have the same F3 and trying to "make it work"
As stated in other threads, you run a very real risk of trying to fix a broken $20 speaker by using $50 worth of parts and $1000 worth of time. 
If it's for fun, go for it. If it's for practical utility, you should've thrown away the old speaker and bought a new one. 

Crinacle ranks the SRH840A above the LCD2C. Rather than being 4x as good in general, it's more accurate to say that most of the differences above ~$200 tier are more a matter of different preferences and tradeoffs than strictly being better.
 
For what it's worth, if I could only own one and they were the same price, I'd rather have the HD600 than the LCD2C. The LCD2C is better in some ways and worse in others, and I value the HD600's advantages more. You may feel differently, and that's entirely valid. If anything, it's a good reminder that trying out headphones in person before making a purchasing decision is the best approach, whenever practical.

The HyperX soundcard will work better than onboard.

To be fair the 2018 Sundara isn't supposed to have more bass than the HD560S.
 

 
Power-wise, here's an approximate comparison:

Source V into 32Ω (Vrms) Power into 40Ω, at 32Ω voltage (mW) Sundara max SPL (dBSPL) FiiO KA3 SE 2.05 113.6 110.5 FiiO KA3 Balanced 2.75 204.4 113.1 Moondrop Dawn 3.5mm 1.6 69.2 108.4 Moondrop Dawn 4.4mm 2.1 119.2 110.7 Generic ALC1220 Motherboard 0.6 9.7 99.9 LG V50 Quad ES9218P 0.34 3.1 94.9  
Sensitivity/Impedance measurements for Sundara taken from ASR.
KA3 measurements from 7AudioLab.
Moondrop Dawn measurements from 7AudioLab. There should be no difference between the output power of the standalone 3.5mm and 4.4mm versions of the Dawn and the combined Dawn Pro, because they use basically the same output circuit. This is supported by the 4.4mm Dawn's measured output power almost perfectly matching the spec on the Dawn Pro.
ALC1220 Motherboard numbers based on Realtek datasheet. The ALC1220 and ALC4080 are interchangeable power-wise since they should share the same analog section.
LG V50 based on my own measurements just now. They are close to ASR's measurements of the G7.

Normally your use case would be ripe for something like the M50X wireless or K371 wireless, since those are meant to be studio headphones with added bluetooth. But in this particular circumstance I'd probably go with a Bose QC35, if the market for those is reasonable where you are. The XM3/XM4/QC45 are also good options.
 
There are a couple of factors here that favor ANC headphones:
Isolation is useful Long hair is likely to breach the headphones' seal which completely changes the bass response of closed-back headphones; feedback ANC microphones correct for this effect Latency isn't an issue The downsides of ANC still remain:
Worse mid and treble timbre (but not necessarily wideband frequency response) due to imperfect correction signals Possible "cabin pressure" discomfort depending on the person Poor perceived soundstage due to internal microphones "correcting" the listener's pinna response Only sound good with ANC on I think the QC35 is a slightly better fit than the other consumer ANC headphones because:
It is close to Harman neutral, with the exception of grainy high treble above ~7kHz The bass correction on the Sonys is underdamped, which means they have more rolloff and unexpected midbass peaking if the seal is compromised (though still not as bad as most passive closed-backs) At least in the US, the Bose models are cheaper Better controls (IMO) The Sony models still have a couple of upsides worth considering:
For most people the Bose is more comfortable due to the reduced clamp force, but the higher clamp force on the Sonys may be a positive for someone with a small head if too little clamp force is an issue with other headphones More consistent high treble, though the bass is boosted and the mids are subdued compared to neutral (though not extreme by any means) Weaker at the hinges but otherwise difficult to damage in a cosmetically obvious way, whereas the Bose metal cups are prone to purely cosmetic denting Better-looking (IMO)  
Overall, I don't think any of the headphones listed above are bad choices, and it really comes down to weighing the benefits of each vs your local pricing.

Which studio headphones are you looking to match the sound of, or are you more interested in a certain neutral reference curve?
 
Are ANC, latency, or passive isolation important? Will she be wearing the headphones over long hair and/or glasses?

The practical answer is;
Upgrading your soundcard (headphone amplifier and DAC) is a marginal, almost pointless, difference in improving sound quality.
You should only upgrade the headphone amplifier if your current headphones require you to turn up the volume to like ~70% for a comfortable volume
You should only upgrade the DAC if the onboard audio is noisy
There are two camps of people regarding the importance of DAC/amps. Let's ignore them since you're not an audiophile.
 
Upgrading your headphones/speakers is the most noticeable way to improve your audio experience.
Since you seem to be keen on picking out audio details for improving your gaming performance, you should look into detailed, open-backed headphones (if your environment allows)
Audio-Technica ATH-AD700X ($118) AKG Pro Audio K702 ($189) Otherwise, the Drop + EPOS PC38X in your original post is also a good option since it's based off Sennheiser's other well-received headphones (I think?).