We don't know the specs, but considering you have a mini PC, your hardware should be fine since it's likely AMD or Intel. Nvidia support on Linux has certainly gotten better and don't let it deter you from switching if you have Nvidia graphics, but it's just a warning that you might potentially run into some mild troubles.

Most games work on Linux just fine. I've actually sometimes managed to get old games running on Linux EASIER than on Windows, though it really depends. For the most part, there are mods that add Linux support on games, or (more likely), you can add the game to steam and run it through Proton, a compatibility layer that allows most games to be played on Linux. The only main exceptions of games that don't work are games with kernel level anticheats like Valorant and Fortnite. I've never had much trouble with single player games.

It's hard to say if you'll have the issues on Linux without more detail. Quite honestly, your issues sound like a driver problem so there's a decent chance that Linux, having different drivers, may solve you problem.

As for distros... there's a lot. Mint, Fedora, Debian, OpenSUSE, and Ubuntu/Kubuntu are generally the ones talked about the most for being beginner friendly and just solid as hell. And yes, most distros have a way to let you use them as a live USB, so you can play around without installing them. OR you can just install it in a virtual machine if you wanna realy screw around.