Stronger notions of uniformizations?
I think the answer to you first question is negative: A circuit has a fixed number of inputs, and thus, IMO, we can only talk about "families" of circuits, rather than just one uniform circuit.

Regarding your second question, you may note that there are "uniform families of circuits," whose description is generated by a Turing machine. That is, let $\{C_n\}$ be a uniform family of circuit, and let $M$ be a Turing machine. Then, for each $n$, $[C_n] = M(1^n)$, where $[C_n]$ denotes the description of $C_n$.

There are several complexity classes below P, defined by uniform families of circuits. For example:

$\mathbf{NC}^i$ is the class of decision problems decidable by uniform boolean circuits with a polynomial number of gates and depth $O(\log^i n)$.