I’m using my old desktop from 2010. There’s no such thing as a server that can “do it all”, but any computer from the last 10 years would probably be a fine place to start. The more you do, the more likely you’ll be to hit some sort of performance limit, and by that time you’ll know more about what you actually want.
In short, find old cheap/free hardware and start playing around.
The obvious answers are the games we endlessly replayed historically: Mario Kart, Goldeneye (VS mode), Halo (VS), Smash Bros.
If you specifically want ones on PC, I’d suggest Starcraft, Age of Empires, and probably Counter Strike (I wasn’t into that one, but it had a huge following).
Many board games fit the bill as well. Codenames (physical or online at horsepaste.com) comes to mind, and another commenter also mentioned chess.
Basically any games that were made before endlessly grinding became a thing (yep, that’s only been a thing for a decade or two).
If you feel most familiar with Debian, I recommend looking at Mint. It has great compatibility and is otherwise easy to use, and it handles GPUs well (including Nvidia). In terms of hardware, I’ll agree with the others here–used is the way to go, you’ll get a lot more for your money than buying new.
That being said, you can also look for clearance and/or refurbished PCs. This one would probably be powerful enough.
Doubtful. Without Linux running the majority of the internet, I doubt the desktop scene would look the same as it does today.
I’m not an expert, but any time I’ve needed to do this, I set up my own router as a client to the parent router, and I set my router (client) as the DMZ in the parent router. Effectively you end up with two routers that are both (more or less) connected directly to the internet, without the two networks messing with each other. It’s also minimally invasive to the parent router (even old stock firmware has always had a DMZ option).
The tricky part then is using the wireless connection as your “WAN port,” rather than a physical one. In which case, as long as you can install OpenWRT on it, you should be fine.
The answer also depends on your level of experience and how much you want to learn doing this. You mentioned you haven’t done this before, but are you otherwise comfortable using computers and figuring things out? Are you familiar with Linux and/or the command line? In addition, are you hoping to tinker around and learn a lot from this, or are you more concerned with just setting it up so you can use it?
There are options for all levels of expertise and technical interest, but I recommend starting with any hardware you already have or can aquire for cheap/free (especially if you’re hoping to tinker and learn more). As another commenter suggested, finding an old desktop or laptop and putting a NAS operating system on it would be a great starting project. Then once you play around with it, you’ll know if/where you want to spend some cash on something better. If you don’t have old PCs laying around, check on whatever you use for local buy & sell listings, you can probably pick up something for pretty cheap.
If you’re mostly looking to play around and you don’t have any extra hardware, you can also try things out in a virtual machine (download VirtualBox), which will let you learn without any monetary investment.