Rootzy

i think you don't need control over hdmi for monitoring your video and if the hdmi output setting is correct and you continue having problems perhaps there is a fault with the hdmi connector on the camera.  can you test with another monitor?

It really does seems like a fault in either the monitor, cable or the actual HDMI port on the body. To test the body maybe if possible, use a different monitor and cable. To test the cable, switch monitors. To test the monitor just plug the HDMI into a laptop or computer. That way you can identify the fault if there is even one at all. 
More often than not though the monitor is at fault but its not uncommon to come across a bad port or cable. 

A used Panasonic G7 is the only mirrorless/dslr you will find that does anything higher than 1080p in that price range. And most cameras doesn’t shoot 1440p. Either you go for 1080p or 4k in most consumer dslr/mirroless. 

It will just draw what it needs. Does not matter if you get a 1 Amp powerbank or a 10 Amp powerbank. 

You don't need to worry, the camera will draw whatever it needs too.

it depends on what you are comparing to what. 
 
For instance, while native glass might be an issue, the ability to adapt glass to mirrorless makes it technically hva more lens options, especially if you don't really care for AF (e.g. if you are doing video in some cases.) 
 
Also, if you can attach the camera to a battery bank you have a better battery life. 
 
The best low light performer is actually the a7siii. I will shoot at insanely high isos. 
 
Similarly: 
 
Better burst modes than what? a 1dx2/7d2/d5/d500? Hell no! 
 
Also, EVF is a point of preference imo its not that great. I like the customisability and ability to use peaking and have a live histogram, but the exposure thing never really got me, I have had a lightmeter for that for ages. 
 
In regards to the Fuji s5 thing, I wouldn't go with one, just due to age and the fact that you are a beginner. 
 
Set aside ~$250-$300 for a good, affordable (NOT CHEAP!) tripod(Benro, 3 legged thing, Mefoto and Manfrotto's higher than the befree range are all good for this), an intervalometer (remote trigger basically, ~10$ off amazon) and a couple of cards. 
 
Then since you are going with a crop sensor, look at a used Sigma or Tamron 17-50mm f/2.8. ~$200
 
That should leave you with just over $500-$600 to play with. You should be able to get a used 750d or d7200 in that budget. I would go with the nikon in this case since the d7200 is considerably better than the 750d. A 70d or 7dmk2 could possibly be workable in that budget as well, and either would probably do better if you want to record video. 

Now are you saying that you can abuse the .app system? 
 
Tsk tsk tsk  

Most of the comments above covered most things but as you mentioned, definitely investing in some lighting would help! Good lighting can definitely improve any camera's quality!

A5100 or A6000 would be my recommendation.

Lumix G7 is great, but I agree with Enderman that an a5100 or a6000 would be a good choice and is smaller than the G7

100 mbps is quite a lot, getting an extra 20-60mbps wouldn't make a significant impact at all.  All it offers is h.264 in MP4 and AVCHD containers with Interframe compression that doesn't require much more data.  Besides, the G7 doesn't have V-Log, so unless you need it for serious color grading there's no point in asking for higher bit rates.