1. Card is idling at 50C-55C, which is kinda high.
As Darren and Beers have been saying, this is totally normal. Even my 980Ti idles at 45 C, 50 C in the summer. Even more so, idle temps don't mean much of anything, so quit focusing on them.
2. When playing Fallout 4 at low quality, the FPS is a steady 60 and the fan is around 30%, but after ~10 minutes in game my GPU temp spikes to around 80C, the fan ramps up to 100%, and my FPS drops to the 40s until the card cools down, then I get 60 again, and this process repeats obviously.
Your card is pretty old for this title, even if it does eat CPU fairly well. This isn't the card breaking, it's the card showing it's age. In a game that stresses it to it's limit, it's going to get hot (like any CPU/GPU should and will), then the fan turns on to cool it off, and in reaching that 80 C, you're getting a tad close to the ceiling in temps for the card, hence the minor throttling you are noticing here. Nothing in this is strange or unexpected. You have an old card, trying to play a new game.
3. In TF2, for the 3 years I've had this PC, TF2 has played perfectly fine... until now. In TF2, the card almost never went above 60C and the fan always stayed silent. Well, now, as soon as I launch the game, my GPU temp spikes to the 70s-80sC and the fan ramps up real fast, yet I still get a solid 60FPS.
Perfectly fine == solid 60 FPS. The card is probably getting a bit old, and is going to heat up easier and more often, you need to accept this, or replace the card.
To my knowledge, the drivers are up to date, and I blow dust out of the card weekly. It is a Sapphire Radeon 6950 2GB from 2010 with stock clocks on it, no OC. When I look at the card in my PC it is noticeably sagging and the little motherboard part of it (the blue thing with the chips and capacitor things) is bending a bit.
It's likely been sagging since the day it was installed, and has gradually gotten more saggy as time went on.
Backplates aren't intended to be a fashion statement, that's simply a convenient side-effect of having a sturdier card.
Nothing you are mentioning here (repeatedly, I might add), is an issue. The card is aging, and therefore getting hotter than it used to. That happens. You aren't going to reasonably fix that. What I suggest, is to listen to Darren. Save your money. Look at parts a bit (
tentatively, don't make any plans, because things will change). When you have the money, then you can start making real parts list and getting help and advice.
In the meantime, make the fan curve more aggressive (for God's sake don't put it at a fixed percentage

), and live with the temps and performance.
We've all (some of us repeatedly) told you in this thread that this
isn't an issue, so please believe us. You're going to give yourself a hernia if you keep worrying about idle temps and mediocre performance on a 5 year old card this much.