Yes, you’re over your head. Backer board is not waterproof. Grout is not waterproof. Tile is not waterproof. A separate waterproofing step is what keeps the mold and rot from invading your walls and floors. That wasn’t done on your project. It MUST be done in the remodel.

If you have time to research, own tools and have some skills, buy a TCNA manual and learn. If you’re going yo have to pay a pro, so the same thing. Research and understand. But in the mean time, you’re going to need to hang plastic up over those walls if you’re going to have to use that bath. If you have another bath, then you use it and leave that alone with the door closed.

https://www.tcnatile.com/products-and-services/publications/218-english-publications/188-handbook/960-2019-tcna-handbook-for-ceramic-glass-and-stone-tile-installation.html

I'm not super worried about the actual process of putting up new insulation, waterproofing the backer and retiling. I know how to do that, I'm more asking about how bad this discoloration/ mold problem is. We just moved into this house a year ago.

Looks like you probably have a leak. I would fix that first and would also remove all insulation that has mold on it. Insulation is pretty cheap, and best to be safe, than sorry. Dont close up that opening, until you know the pipe that is leaking has been repaired and done correctly so it no longer leaks.