My computer is all kinds of screwed up... is my Flash problem solvable?

This sends up a number of red flags for me.

USB Sticks are a common vector for malware.  It's possible to infect the firmware on the actual stick itself, such that the stick can be blank, and it can still infect every machine it's inserted into.  That the symptoms appeared after a couple bad memory sticks, and your machine no longer does important things like installing security updates sends all kinds of red flags for me.

If this was one of my personal machines, I'd back up all of my important data to an external hard disk or online service (google drive, dropbox, etc.), format the machine and reinstall Windows, fully update it, then reinstall my software from pristine sources.  Most Internet Service Providers in the US give you a free antivirus license with your subscription, and Microsoft Defender is a free option.  I'd also probably retire the USB sticks (with a hammer) and switch to using a cloud service for transferring files between machines.

There's still a remote chance that sophisticated malware might survive by infecting firmware on the machine, but given that it's older hardware, a wipe and reinstall is as thorough as you're going to get.  It's a wasted afternoon, but far less hassle and worry than identity theft, particularly if you're using the machine for sensitive things like banking or healthcare.

That said, if you don't want to, or can't go that far, you could try just removing and reinstalling Chrome.

You can also go to the following location in the address bar:

chrome://components

You should see Flash Player listed there and attempt to update it.  It takes a few minutes for Chrome to download and install Flash after a new install, but it should happen automatically in the background.

If you're not reinstalling, you might also want to check the disk for errors and defrag it:

Check your hard disk for errors - Windows 7:

https://support.microsoft.com/en-us/kb/2641432