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Note that the appropriate notion of morphisms in $CBA(M)$ are ($M$-)complete embeddings and generic ultrafilters correspond to ($M$-)complete embeddings homomorphisms of $B$ onto $2$ with the caveat that these do not exist in $M$ when $B$ is non-atomic. (Though, as Joel pointed out, we can consider Boolean ultrapowers instead of generic extensions and relax the completeness requirements.)

  1. Yes and no. As described in this answer, if $G$ is a generic filter over $B$ and $M \subseteq N \subseteq M[G]$ is an intermediate model that is generated by a subset of $M$ in $M[G]$ then $N = M[G_0]$ where $G_0 = G \cap B_0$ and $B_0$ is a complete subalgebra of $B$ in $M$. When $M$ is a model of ZFC, then the intermediate models $N$ that are generated by a subset of $M$ are precisely the intermediate models of ZFC, so this does not always classify all the intermediate models of ZF.

  2. No. There are nontrivial complete Boolean algebras that have no nontrivial automorphisms but plenty of complete subalgebras (see references from Jech and Shelah, Simple Complete Boolean Algebras, arXiv:math.LO/0406438). In general, complete automorphisms of $B$ control which elements of $M[G]$ are definable using parameters from $M$: the more automorphisms there are the fewer elements are definable.

  3. Change of base model is always difficult because complete Boolean algebras and complete embeddings homomorphisms may fail to be complete in a larger universe and may fail to exist entirely in a smaller universe. One can always complete a Boolean algebra in a larger universe, but once an embedding a homomorphism fails to be complete it cannot be "corrected" to a complete embeddinghomomorphism. (For example, the complete embedding homomorphism $e:B\to 2$ corresponding to a generic ultrafilter $G$ on an atomless Boolean algebra $B$ fails to be complete in any universe that contains $e$ since $e(\bigwedge_{x \in G} x) = e(0) = 0$ but $\bigwedge_{x \in G} e(x) = \bigwedge_{x \in G} 1 = 1$.)

A good starting point for intermediate submodels of forcing extensions is Grigorieff's classic paper Intermediate Submodels and Generic Extensions in Set Theory, Annals of Mathematics 101 (1975), 447–490.
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Note that the appropriate notion of morphisms in $CBA(M)$ are ($M$-)complete embeddings and generic ultrafilters correspond to ($M$-)complete embeddings $B$ onto $2$ with the caveat that these do not exist in $M$ when $B$ is non-atomic. (Though, as Joel pointed out, we can consider Boolean ultrapowers instead of generic extensions and relax the completeness requirements.)

  1. Yes and no. As described in this answer, if $G$ is a generic filter over $B$ and $M \subseteq N \subseteq M[G]$ is an intermediate model that is generated by a subset of $M$ in $M[G]$ then $N = M[G_0]$ where $G_0 = G \cap B_0$ and $B_0$ is a complete subalgebra of $B$ in $M$. When $M$ is a model of ZFC, then the intermediate models $N$ that are generated by a subset of $M$ are precisely the intermediate models of ZFC, so this does not always classify all the intermediate models of ZF.

  2. No. There are nontrivial complete Boolean algebras that have no nontrivial automorphisms but plenty of complete subalgebras (see references from Jech and Shelah, Simple Complete Boolean Algebras, arXiv:math.LO/0406438). In general, complete automorphisms of $B$ control which elements of $M[G]$ are definable using parameters from $M$: the more automorphisms there are the fewer elements are definable.

  3. Change of base model is always difficult because complete Boolean algebras and complete embeddings may fail to be complete in a larger universe and may fail to exist entirely in a smaller universe. One can always complete a Boolean algebra in a larger universe, but once an embedding fails to be complete it cannot be "corrected" to a complete embedding. (For example, the complete embedding $e:B\to 2$ corresponding to a generic ultrafilter $G$ on an atomless Boolean algebra $B$ fails to be complete in any universe that contains $e$ since $e(\bigwedge_{x \in G} x) = e(0) = 0$ but $\bigwedge_{x \in G} e(x) = \bigwedge_{x \in G} 1 = 1$.)

A good starting point for intermediate submodels of forcing extensions is Grigorieff's classic paper Intermediate Submodels and Generic Extensions in Set Theory, Annals of Mathematics 101 (1975), 447–490.