In what cases should I completely state a theorem I'm about to use?

For my taste, I'd "recall" the precise statement exactly when you need it, in the internals of your proof, as you seem to indicate. I myself have become ever fonder of a math writing style which does not require so much flipping back-and-forth to understand what's being said. (Especially the otherwise-precise quasi-Bourbaki of referring to things by some (necessarily artificial and meaningless) numbering scheme, rather than any sort of descriptive reference.)

In particular, for simply the statement (rather than proof) of a result, adding an appendix would make things harder to read for many people, and the people who already know the result would not gain much. Skipping over known things is easier than flipping back-and-forth.

That is, allowing your readers to read straight through seems to me the ideal. So, no, similarly, don't introduce all the notation at the beginning and then expect people to remember it. Sure, you could have an appendix for reference for notation, but it really should be explained when first used, ... in my opinion. That kind of thing.

Consider having an appendix/annex to the thesis where you state things that are essential to your argument and which your reader may want to see without looking elsewhere. For some things, perhaps in this case as well, a footnote citation of the lemma, including its name, might be enough since most people working in probability will know of it by name. Then, as a step in proving the current lemma/theorem, just refer to the footnote or appendix entry. The reader can take a detour if they feel it necessary.

But, you are correct not to make it hard to follow the flow of a proof.

A full statement of such a thing might be necessary if you were proving a variation or an extension of that thing. And in that case it might be necessary to work the statement and even, perhaps, a proof outline of the original into the text itself so it is clear how you are extending/modifying.