Long story short, the (attached) garage on my house has foundation issues and also structural issues with the roof. (The main structure does not have these issues.)
I know I need to do something about it. But I don't know what to do - Can it be repaired? Do I need to tear it down? Something else?
Which profession to I hire to... help me find the correct solution to this? A general contractor? structural engineer? architect? something else?
(I'm in the U.S.)
Call a few General contractors and get their opinion on what need to be done.
They will know if something is a normal repair or if engineering needs to be consulted.
Get more than a few quotes. Don't be surprised if some say they aren't interested or can't help you. Don't assume that the highest bid is from a GC trying to "rip you off". Take you own notes. Find out how long the project will take. When will it be able to be started. Ask for referrals. Don't be afraid to ask if the GC has done this kind of work before.
Get the proposals and compare what each plans to do to correct your issues. Then you can ask more questions here. We do not debate on pricing, however we can advise on material and methods.
This isn't an exact answer but will lead to your answer.
Have reputable companies come out to bid on fixing all of the issues. You are probably working with companies that do a lot of add-ons but if you are in a larger populated area there are probably 4-5 companies that specialize in garages.
Have them come in and give you quotes on everything. Note that if there are things like concrete work they will probably contract that out - but they are contracting that out to a concrete service they have a relationship with. While getting quotes it is pretty easy to deduce who knows their stuff, who is trying to gouge you, who missed easily identifiable issues.
For example if I bought your house knowing about the garage I would have a few guys/companies come out and bid and I would say something like "Hey this garage seems to have some roof and foundation issues but I want this thing 100% ready so add on anything else you see that is problematic."
Two things going on here. First there are issues. If there are two issues you notice there might be three more you don't. Second if you are having work done and let's say you would like everything drywalled and primed. It is often much cheaper done together than separate (for a home owner).
So the best answer is a construction company that specializes in garages (for instance if you need roofing structure they may have that already in their supply chain). And they probably have 5-10 guys at least that have seen all your problems and possibly even fixed problems from the same builder who did yours.
