Reconshire

No
 
But is the cooler new? It says in the manual it comes with paste.

Please read the installation manual. The insulation pad is for separating the backplate and motherboard to prevent shorting. 
It's not the same as a thermal pad and insulation is the opposite of what thermal pads / paste do. 
You might be able to run the system with no thermal paste and just dry mount the cooler onto the cpu for a windows installation but the pump and fans might  have to run at pretty high rpm to keep temps in check. Make sure there isn't any auto-oc'ing or enhanced turbo from the motherboard settings if you decide to do this.

Thermal paste isn't meant to insulate the cooler from the CPU..

I got it working, I just booted it from the usb and installing windows rn. THANK YOU SO MUCH FOR ALL THE HELP 

its pins are smaller.
 

You don't really need them. I never connect mine because the led's are annoying af, especially the hdd one. 
 
If you have close up shots of the board we might find them tho. 

The other three are apparently for a tri-color led. I've read that it might give you errors of those are not connected but that shouldn't be an issue. 
 
https://www.dell.com/community/Optiplex-Desktops/Optiplex-5-pin-power-button-assembly-pinout/td-p/6234979

Also what resolution/settings?

I mean, this is a mid range gaming rig... at best. You can't expect tons of frames if you're playing above 1080p. Plus you only have single channel memory, so that certainly doesn't help things. 

Sadly, I could not find any details in the H310M-K User Manual.
 
Some Asus motherboards include a labeled plastic block to which the case front panel leads connect. The block then plugs into the motherboard front panel header. If memory serves this is called a quick connect. If not present, see the link in Best Answer of http://www.tomshardware.com/answers/id-3750348/asus-prime-h310m-front-panel-connectors.html.
 

Do you see the white text beside the connector that tells you what to plug in where?

The motherboard is labelled in the bottom right corner with tiny words that say where they go.

Also, did you know that motherboard instruction manuals are available for free on the internet from the motherboard website?
https://www.asus.com/ca-en/Motherboards/PRIME-H310M-K/HelpDesk/ 

No problem. Have fun with it!

Laptop speakers.

If you can find a header labelled P6 or SPKR, that should be for the speaker (black & red wires). (This speaker will typically emit BIOS beeps including beep codes if the POST encounters any errors.) See section 7.6 of http://h10032.www1.hp.com/ctg/Manual/c01960471. According to pg 108 of http://h20628.www2.hp.com/km-ext/kmcsdirect/emr_na-c01941953-2.pdf the motherboard connector should be white. Page 169 details steps.
 
No idea what the other, two part piece is. They look like switches that mount to the chassis for front panel buttons on the face plate. Possibly power and reset?
 
 
 
 
 
 

For $30 it's okay, not great, I wouldn't pay more than $20. You can get a new Kingston A400 for $35aus and a 240GB for $45. 120GB fills up pretty quick as a boot drive so if it was me I'd get the 240GB SSD for $45.
https://au.pcpartpicker.com/product/btDzK8/kingston-a400-240gb-25-solid-state-drive-sa400s37240g

I mean yeah $30 sure for $40-50 you could get a new one with more capacity but thats up to you.

Yeah it's all compatible but your going to experience a bottle neck with the pentium

That's gonna bottleneck like hell. Go with a Ryzen 5 1600 if you can find one and a B350 (MSI or ASRock if you can). If you can't, any Ryzen will be better, even the R3 2200G. Keep in mind it pretty much requires dual-channel memory and likes faster memory, 3000MHz is the sweetspot, but 2666MHz will do. AM4 has MUCH greater upgradeability.

Why such pricey case in a build with dual core pentium? It costs around same as cpu. Buy ryzen, even ryzen  3 1200 is better than pentium dual core. If you can't pay more than you would pay with pentium, then buy cheaper case and different psu and you are set. 80 + gold psu is nice, but bronze is enough with this budget build.

CX450M

I would go with a grey-label CX (2017) model. The 500B is pretty bad