Edit: I probably should have mentioned that I am quite familiar with permutation models of ZFA+embedding theorems and transfer theorems (Jech-Sochor, Pincus' theorem) as well with symmetric extensions.
I am not looking for ways to develop forcing extensions of ZF without the axiom of choice; rather I am looking for theorems such as c.c.c forcing does not collapse cardinals and similar theorems extended to the choiceless contexts if possible, or the strength of choice needed for these theorems to hold.
Consider two examples:
Suppose a model of ZF in which the axiom of choice does not hold. Can we, by set forcing add the axiom of choice? If not, can it be done using a machinery similar to a symmetric extension? If we can in fact find such extension, does that mean the model without choice is a symmetric extension between two larger models?
Suppose A is an infinite Dedekind-finite set, what can we say on a forcing poset based on A (either domain of functions are partial to A or the range is in A)? Can we "collapse" amorphous sets onto ordinals? Can we collapse one amorphous set onto another? And so on.