Actually wait: I've misunderstood the question again, and what I said doesn't necessarily apply. (It applies when X comes from a finite object in the "genuine" equivariant category. But your finiteness condition looks like it might be weaker than that, even without allowing retracts.)

Ah, misunderstood you. I don't have very much helpful to suggest, then. One comment: in the special case G = Z/p, if a space X with an action of G is to have the form you describe, then it satisfies the Sullivan conjecture. In particular, the homotopy fixed points for the action of G on the p-adic completion X_p must itself arise as the p-adic completion of a finite space. Possibly this could give you a way to test if something is a counterexample? (A finite X w/G action such that (X_p)^hG is not the p-adic completion of anything finite?)

In Fun(BG,S), the orbits G/H generally won't be compact when H is nontrivial. But G itself generates the compact objects under finite colimits and retracts (this follows formally from the fact that it is a compact object which corepresents a conservative functor: in this case, the functor which forgets the G-action).

Let I be the fiber of the map A -> B. Then the fiber of the map A -> Tot^n is the (n+1)st smash power of I over A. For these fibers to vanish in the limit, it suffices that they are getting more connected. Since A is connective, it suffices for I to be connected, which is immediate from your hypothesis.

Any line bundle over the spectrum of a local ring is trivial.

For a counterexample in spaces, let X_* be the simplicial set with vertices the integers and edges joining each integer n to n+1. Regard X_* as a simplicial object of spaces whose geometric realization is contractible. Then a product of infinitely many copies of |X_| is contractible. But the geometric realization of a product of infinitely many copies of X_ is not connected: for example, there's no path joining the identity map Z -> Z with the constant map joining Z -> {0} -> Z. (Contradicting your "elementary computation".)