I don't have a specialized knowledge about regularizations, but as the first direction I think:
(1) yes, it's an identity if you look into definition of zeta-function (I thought of a different definition of zeta-regulaization)
(2) not sure, I'll check later. But it seems to be an identity as well.
(4, 5) TR only depens on eigenvalues. I don't remember what the status of eigen{AB} = eigen{BA} is, but I think its true under standard conditions on A and B -- if they are self-conjugate in a Hilbert space.
Let's say you try to compare regularized \det(1-AB) and \det(1-BA). I think the standard reasoning applies and says that those are equal; same might apply to (A+B).