BSOD from NTOSKRNL.EXE, 0x00000101, even after reinstalling OS

I've heard good and bad things about that Catalyst. Just install the basic drivers for that video card. I'm pretty sure you didn't OC the CPU on accident :).
Did you go into your current BIOS and set it all back to default? If so, that's a good step.
After you have done that part. Look for your "clear CMOS" button. Refer to the motherboards manual to find this button. DO NOT press that button unless your computer is powered OFF.
Once you have successfully understood the flashing (updating) process for that board, then attempt it. As I said earlier.. know your BIOS and know it well. It really pays off

I seemed to have done something wrong with updating the BIOS. I didn't damage anything, it's just when I ran the installer, nothing happened and my computer froze.

Is updating the BIOS absolutely necessary, do you think? I'd rather not risk break something because, even though I'm getting BSODs and occasional freezes, I'd rather have a computer that works but freezes occasionally than one that doesn't work at all, at least until I can afford new parts if necessary.

So would it be possible to continue troubleshooting and assume the BIOS aren't causing any issues, or do you think these outdated BIOS are absolutely a problem?

Just now my PC crashed but instead of going to a BSOD, the display just garbled up and I had to power off my PC. I was just using a web browser and a few other inconsequential programs, such as this thing and Tweetdeck.

This has happened twice in 24 hours, actually, and this is the fifth or so time it's happened in the past few weeks. I got a photo of it the first time it happened and I've attached it to this post. I didn't post about it the first time because I figured it was just a fluke.

Probably safe to say this is related to my other problems?

Alright, something astoundingly weird is happening.

I was using the free trial of Malwarebytes Anti-Malware and it expired yesterday, so every so often a little text bubble would appear telling me my trial was over and to click the bubble to see what to do about it. Clicking it brings up the screenshot of the window attached.

Clicking on the third option, "End Trial", makes my PC crash in the same way as my last post--screen garbled, have to shut it off manually. And it's extremely consistent. I've clicked it three times and each time the same crash. It doesn't matter what programs were running, either. The first time I had a browser with a livestream open, the second time some Windows startup programs were opening up, the third time I'd given my PC enough time to get all the startup programs running and everything.

So like I said, extremely weird, but hopefully whatever MBAM is doing to consistently crash my PC can help us more easily diagnose the root cause of my crashes, maybe? I've made a restore point just in case clicking on "End Trial" actually works one of these times.

Edit: After making this post I clicked on "End Trial" a fourth time just to make sure this wasn't three consecutive flukes, and no, it's not. It did it again.

I don't mean to keep posting in a row like this but the crashes are getting more frequent. Now in the past 24 hours my PC has frozen twice, similarly to the way it did in post 13, only instead of the screen getting scrambled, everything just halted entirely.

I wasn't really running any programs I hadn't been able to use before without any issues, just a web browser.

I haven't actually had any more BSODs, it's all scrambled screen crashes and just complete halts. I'll try removing the GPU.

I came down with the flu and was unable to post until now.

I cleaned the dust out of my PC and unplugged the graphics card, but when I turned on the PC, I didnt receive a display signal. Is there something I have to do to switch to my APU's integrated graphics?

I think it's safe to rule out the graphics card being a problem. Last night I was able to run Steam, a VOIP client, and a game for four hours. Today, I was using Steam and a low-end game, but I had Firefox downloading something in the background. Firefox had one of its famous memory leaks and pow. Freeze.