How to deal with results that are given as exercises

In mathematics, if you cite a result, you should really be able provide the proof if somebody asks for it. So I would strongly recommend against option 3.

One thing you might consider is writing the author of the textbook, explaining your situation, and asking if he's willing to send you a proof. Then you can cite it as

problem 8.35 in textbook, proof provided by private communication with [author]

even if he would rather not have a proof of the exercise published.

Of course, if he replies "I think I had a proof when I wrote the textbook, but I've forgotten it now," you might need to come up with a proof in some other way.

There is at least one exercise in a textbook whose proof is highly non-trivial, and was an open problem for several years before a journal paper giving the proof was published (and the only indication in the textbook of this is that part (b) of the problem is "optional").

Don't give any citation at all and assume these concepts are well known to anyone working in the field? - I wouldn't give the definition of a vector space in my thesis, so this might be justifiable. However, I wouldn't consider those theorems as basic as the definition of vector spaces...

The general idea for citations is to include a reference is if you would not assume a typical reader of your thesis/paper to know these results. Think: will this reference be useful for someone reading this? For a thesis, one is often a little more liberal with background and references than a paper. Since you yourself don't seem to know (proofs of) these results, I would include a reference, though of course it's your choice.

Do a heavy amount of googling to find sone sources which prove those theorems? - This might take a considerable amount of time, so I'd like to avoid it. It might also turn out to be impossible, since the phrasing might be different enough in the original sources so that I just won't find them

I think this is wrong attitude to take. I understand you may be under time pressure, but you should try to understand everything in your thesis as well as you can, including related work. It's often not feasible to understand proofs of every result you mention (maybe you will many years later), but you should at least know references. Whenever I write a paper (or writing course notes) I spend a long time reviewing literature. In addition to making you feel more comfortable about what you're writing, being familiar with the literature is important for your mathematical education.

What I would do is spend an afternoon skimming through other books/surveys on the field. If it really is something rather basic, there should be another book that goes through this. If that doesn't work, you can try to prove the exercises on your own as well as ask your advisor if s/he knows a reference where these things are proved.

Cite the Problem from the book I'm working with? - I don't think I can do this, since the results aren't proved in the reference I would give

You can, but I agree it's generally nicer to the reader to provide a reference that has an explicit proof.

All of these questions you asked are things that you should be able to ask your advisor, though it's good if you can show some independence by being able to survey the literature on your own.

If it's a fact that needs to be justified but is not relevant enough to devote space in your thesis to proving it in detail, it's better to cite a source that gives the proof, if you can find one. If you can't, I recommend trying to come up with a proof of your own, and writing in your thesis something like

This fact can be found as an exercise in [Textbook, problem 3.14] and can be shown by [1-2 sentence description of the proof strategy].