My PC is custom-built to be perfect for modern gaming and it runs most big games from nowadays fine, but even with its 3,40 GHz Core i7 proccessor and NVidia GT 620 graphics card I find it to be a bit... faulty, as I can run a moderately-intensive game for about fifteen minutes before it crashes.

It's not the usual Windows BSOD crash, though: the screen goes dark and the music/sound effects glitch up in a very, VERY weird manner, and then I have to hard reset the machine by turning it off and on again by holding the power button down, with everything being fine afterwards. Some less-intensive games take more time to crash, and the lightest ones (emulators, most indies) don't crash at all. (Ikaruga on Steam also falls into the latter category, strangely enough)

I've already tried to solve the problem via guesswork by formatting my rig (I needed to delete a lot of stuff from it, anyway), but even then this annoyance keeps tormenting me and my games. Nowadays I think it's being caused by a faulty or weak cooler (the way the case is configured makes it hard to clean, so maybe it's excessively dusty?) or an outdated graphics card (NVidia says mine doesn't support SHIELD streaming and ShadowPlay, so I guess it's outdated?), but a computer-savvy friend of mine said it's probably a proccessor-related problem (which I sincerely don't think it is)...

So can you guys and gals help me out here? Even though I'm more of a retro gamer, I'd also like to play more modern stuff without worries...

Thanks in advance!

There are a few things I'd suggest:

First, I'd open the case to see your machine is dusty. If it is, get a regular ol toothbrush and GENTLY brush off the inside

If it's not dusty & your rig is prebuilt (with an i7 and a GT 620, I'm guessing it is), I'd suggest calling your manufacturer

If it's a custom rig, I'd call your parts manufacturers one by one until you solve the problem. I'd start w/Intel, then Nvidia, then whoever makes your mobo. Doesn't hurt to go down the chain

Hope this helps, man

Check windows event log.

Check drivers are all functional.

Download and run speccy. It displays hardware temperatures, operating speeds yafa yada.
Play a game and check speccy every 5 minutes.

Is your psu good enough to cover the gpu at full throttle?

just reread your post. It absolutely sounds like overheating. Being dusty would absolutely contribute to increased heat. Check your PC and fans to see if they're clogged. If not, I would run your PC with the case open to see if maybe one of the coolers isn't working properly. That seems like the most likely culprit.

Also, custom built with an i7 and a GT 620? In case you build another PC at any point in the future, the i7 is overkill for literally everything but video processing. i5 is the golden standard

I'd also follow number's advice and download speccy. It's really useful

I 100% agree, overheating is more than likely the issue.  If you have a small form factor case I would recommend purchasing a mid tower case and moving your components over, if you have a mid or larger case I would be looking at your fan arrangement.  Your cpu and gpu fans are blowing air into their heatsinks so make sure each have sufficient cool air intake.  Also make sure your case has sufficient air intake and sufficient extraction.  The gt620 can play modern games at OK settings but a common issue not everyone considers is your using that gpu to its max, I would be upgrading to at least an R7 260 or GTX 560.  

pollution_skunk :  What PSU are you using?  I don't think the issue is dust.  It has to be seriously dusty inside before it will cause any problems.  I had a similar problem a few years ago.  My case was to small which caused excessive heat build up, so I took one of my side panels off as a temporary solution until I got a bigger case with much better air flow.  Taking the side panel off worked like a charm.  Just try it and see how the PC reacts before upgrading to a bigger case.  If the rig still act up then it's something else causing the problem.  Also check your wiring inside that it doesn't go all over the place.  A stray wire can easily jam a fan.  Take some time with cable management and tie them down properly.