Single theorem adaptive vs non-adaptive non-interactive zero-knowledge proofs
It seems to me that the term "adaptive NIZK" is unclear. Where did you encounter it? If you give me the context, I can probably tell you what it was suppose to mean in this context.

Specifically, adaptive NIZK could refer to:


NIZK with adaptive soundness, where the malicious prover first receives the crs as input, then produces a word and a proof, and wins if the word is not in the language yet the proof is accepted;
NIZK with adaptive zero-knowledge, which matches what you mentioned: the simulator is divided into two parts, and first return a simulated crs, then outputs simulated proofs;
NIZK secure against adaptive adversaries in the UC model, which is more complex.


Assuming you meant "adaptive zero-knowledge", I don't think there is any generic way of transforming non-adaptive NIZKs into adaptive NIZKs; in fact, it's an open problem to build adaptive perfect NIZK arguments from standard assumption (those we have are from non-standard "knowledge-of-exponent-type" assumptions, or satisfy only a weaker property, called adaptive culpable soundness).

For your third question, I don't know if there are group signatures that use non-adaptive NIZKs.

For the last question, in a single-theorem NIZK, the zero-knowledge property is only guaranteed for provers that send a single proof after the crs is generated - to perform more proofs, new crs must be generated. Put otherwise, the length of the crs must be proportional to a bound on the number of proofs that will be executed with this crs. In contrast, in a multi-theorem NIZK, the same crs can be reused an arbitrary (polynomial) number of times for generating proofs. There is a generic transformation from single-theorem to multi-theorem NIZKs that uses only one-way functions, see this paper.