Do we have any nontrivial uniform circuits?
Here are answers to your last two questions.

(5) Sorting networks are uniform circuits that sort as fast as the best RAM algorithms, but are definitely not just conversions of RAM algorithms (e.g. quicksort). [AKS83,G14]

(4) Yes, for any $s(n)=(1+\varepsilon) \cdot 2^n/n$ with $\varepsilon&gt;0$, but for a silly reason: Every function is computed by a circuit of size $(1+o(1)) \cdot 2^n/n$. (Shannon proved this up to a constant and Lupanov got the optimal constant.) By the time hierarchy theorem, there exists a function $f$ with uniform time complexity between $\Omega(3^n)$ and $O(n \cdot 3^n)$. This gives a counterexample: $f$ has circuits of size $O(2^n/n)$ (which I think are computable in $2^{\mathrm{poly}(n)}$ time) but is not computable in $\tilde{O}(2^n/n)$ time. You should probably ask for $s(n) = o(2^n/n)$.

This is an interesting question; I hope someone can answer (1)-(3).