Small circuits for circuit evaluation problem
I have learned from talking to Ryan Williams (who deserves the credit for my being able to post this answer) that it is known from Paul and Pippenger that Circuit Eval can be decided by a quasilinear time multitape TM and also that there are reductions from multitape TMs to circuits which give only a quasilinear size blowup. That is, Circuit Eval has circuits of size $(n+s)\log^{O(1)}(n+s)$, as per your formulation.

There is a proof of this here on page 6 (see Theorem 3.1 (Folklore)).