You should clarify the meaning of almost equivalent, since you are quantifying over $V$-generic filters $G$ and $H$, but of course, in nontrivial cases there aren't any such filters (that is, there are no such filters in $V$). So what do you mean? You should express the definition in a way that makes sense inside ZFC. Probably what you mean is that below every condition in $\mathbb{P}$ there is a cone that is forcing equivalent to a cone in $\mathbb{Q}$ and vice versa.

Relevant background: jdh.hamkins.org/…

I want just to use the subspace topology inside the symmetries of $F$. That is, view $F$ as discreet, and basic open sets of permutations is determined by finitely much information about how it acts.

I just meant that if an automorphism (other than conjugation) is nontrivial, then it is moving transcendentals around, but if a sequence of automorphisms are increasingly in agreement on them and their preimages, then the limit will also be automorphism. Can we generate new automorphisms from a given nontrivial automorphism?

Perhaps a strategy for a negative answer might be to try and show the automorphism group, if infinite, is perfect in a suitable sense and therefore size at least continuum?

Wow, this is very nice!

If you have extensionality, then with reflection you are going to get adjunction and therefore arithmetic, and then a form of second-order arithmetic by reflection. Without extensionality, you are going to have to go through the interpretation of set theory in set theory without extensionality, which is already complicated in ZF-ext, but in your weak theory, this will be even harder to verify. I expect it probably works.

Could you clarify whether in axiom 2 you allow $\phi$ to have free variables, and then you take the universal closure, or are you only asserting instances of reflection for sentences? Also, could you explain why you do not include extensionality and other ordinary axioms of set theory? After all, reflection is equivalent to replacement over the other axioms of ZF, but your theory is weak in a way that is irritating and seems pointless without explanation.

You can make a non-computable set as sparse as you like.

The first-order theory of real-closed fields is equivalently axiomatized without the odd-degree polynomial axioms, but instead including the scheme asserting that every definable nonempty bounded set has a least upper bound. That is, definable in the language of ordered fields. So real-closed fields satisfy the definable LUB property, which seems highly relevant for any discussion of completeness.