When citing a theorem/lemma/prop/coro... from a book/article/research paper for writing an article etc. do I have the freedom to:
1. Change notation or introduce my own notation
e.g. "Let $g$ and $h$ be homeomorphism...", but suppose $g$ is already used extensively elsewhere, so can I change it to $f$ and $h$ instead?
e.g. "Let $F^T(-i\omega)$ be the Hermitian transpose of $F(i\omega) \in \mathbb{C}^{n\times n}$", can I instead denote $F^\dagger$?
2. Minor change to the wording (of course without changing underlying logic)
e.g. "Suppose $f$ is holomorphic, then we claim $f$ satisfies $Q$", can I change to to Let $f$ be holomorphic, then we claim $f$ satisfies $Q$?
How much freedom do I have to make these changes when citing a theorem from a book?
I am not a lawyer; go see a lawyer if you need actual legal advice. With that said...
As a general rule, you can always paraphrase something using your own words, as long as you cite and give credit appropriately, and as long as you don't use quotation marks or otherwise indicate that the wording is an exact copy of the original (this is if it's not a copy, of course). The paraphrase should be different enough that nobody would accuse you of plagiarism. [In the first sentence here, "always" means with respect to copyright law — there may be other restrictions due to trade secrets, libel, etc., but they're unlikely to come up in an academic context.]
Needless to say, for a mathematical theorem or the like, you need to be sure that the rephrasing has precisely the same meaning as the original, so that you aren't attributing to the earlier author something with a different meaning from what they claimed.
There's also the question as to whether something is an exact quote (in which case you should indicate that by using quotation marks or the equivalent) or a paraphrase (without quotation marks, so you should be careful not to use the original wording, to avoid committing plagiarism).
This brings up the point that it can sometimes be difficult in mathematics to paraphrase something using entirely different words; there may not be that many ways to say the same precise technical thing. But certainly a descriptive or explanatory sentence in English should not be copied verbatim without quoting it.
Again, I am not a lawyer, but my understanding is that facts cannot be copyrighted, so if there is basically only one way to say something (for example, a formula), it's not subject to copyright. On the other hand, a specific collection or compilation is subject to copyright even if the individual pieces are not; so, for example, it may be infringing to copy an entire table of integrals, because there was intellectual effort and choice involved in selecting which integrals to include, and in what order, even though any one of the formulas by itself might be usable without being infringing.
Entirely separate from the above considerations, you might be able to claim "fair use" in some cases, depending on how the item is being used. But you still need to satisfy normal academic (and common-courtesy) requirements for attribution.