1. What about my OS? I just realized last night that I don't have any possible way to back up my copy of Windows. My computer came with Windows 8 and I upgraded to 10 through Windows' eligibility program, so I don't have it. What am I going to do about that?

You don't WANT to backup Windows, it sounds like something in Windows is what caused your problem. You will need to reinstall it though. To do that you will need to get an installer disc (you can download the file online and burn it on to a DVD [or put it on a flash drive]). You will also need your Windows serial number (I'm not familiar with how to get that on W10, but you can google around until you find an answer).
If any of this is too complicated, you should be able to call Microsoft and they will walk you through it. Or bring it to a Windows store if you prefer. If you aren't comfortable blowing away all your stuff yourself (and it sounds like you might not be) then you should probably call Microsoft and have them help you.

2. There are also a handful of programs that came installed with the computer--the ones I'm specifically thinking about are Microsoft Office and Windows Media Player. You know how tight-lipped and stingy these people are with their products. What do I do about those?

WMP will be on the new Windows you install. My copies of Office came with licence codes, presumably yours did to? Again, googling for how to find those license codes could solve your problem, or calling Microsoft would work too.

3. Is there any way I can POSSIBLY prevent this from happening again?! I'm sick and tired of not being able to use my computer properly.

Without knowing the cause it's unfortunately really hard to say. Someone else might have better suggestions but I would just recommend following good practices while using the computer. Make sure you scan for viruses/malware every couple weeks, avoid sketchy sites, make sure things like Windows updates succeed (ie. don't force-shutdown in the middle of one, etc.), make Restore Points (they could have a different name in Win10) every couple weeks, NEVER move the computer when actively using it (this can harm your hard drive over time).

Come to think of it, did you ever run any tests on your hard drive? Portions of it could be corrupt which could contribute to your issues.

1. What about my OS? I just realized last night that I don't have any possible way to back up my copy of Windows. My computer came with Windows 8 and I upgraded to 10 through Windows' eligibility program, so I don't have it. What am I going to do about that?

2. There are also a handful of programs that came installed with the computer--the ones I'm specifically thinking about are Microsoft Office and Windows Media Player. You know how tight-lipped and stingy these people are with their products. What do I do about those?

You can use something like Jellybean Keyfinder to view the license keys for any installed MS products. If you're sticking with Windows 10, there's a way to clean install it without having to install Windows 8 and then upgrade again. The current method is fairly convoluted, but they're supposed to be in the process of making it so you can activate Windows 10 with the same key that you used to activate your original OS. This might already be the case. If so, you can just download Microsoft's Media Creation Tool, set up a flash drive for a super-speedy install, and you'll be good to go.

3. Is there any way I can POSSIBLY prevent this from happening again?! I'm sick and tired of not being able to use my computer properly.

This trumps everything I mentioned above. Before you think about reinstalling Windows, check your hard drive. Use something like the hard disk diagnostic tools on the Ultimate Boot CD and see if the drive itself is failing. A few random read/write errors can make even a clean OS install go wonky pretty quickly. It sucks to put in all of the work to get it up and running and then watch the whole thing crash the next day :(

-Disciple