1) Is there a description of uniformity just in terms of circuits?
[This is an edited version of my reply to the same question you asked on Dick Lipton's blog. Caveat: I'm not an expert.]
Yes (I think), of at least two different kinds:
a) The circuits are generatable by a Turing machine in polynomial time in the problem input size (as mentioned in some other replies).
(I think this is the standard definition of the concept.)
This covers any circuit family that we could want to call uniform, but as a definition of the concept of P-time, it just reduces the definition on circuit families to the definition on Turing machines, which might not be what you want.
b) If there is a 1-dimensional cellular automaton which evolves the problem input to the problem solution (for a decision problem, the solution would be a single bit in a specified cell relative to the cells containing the input, which is a stable state of the CA), in polynomial time in input size, then this corresponds to a circuit which is periodic in 2D in a simple way (one repeat unit per cell per time-unit), and whose state only matters in a quadratically large region relative to the solution time.
This is a very special kind of uniform circuit family, but sufficient to solve all problems in P, since a Turing machine can be easily encoded as a 1D CA. (This also appears to satisfy the definition of DLOGTIME-uniformity mentioned in an earlier reply.)
(This is similar to the encodings of Turing machines as circuits mentioned in Gowers’ replies on Lipton's blog — in fact, one of them is probably identical.)
One way to encode a Turing machine as a 1D CA: in each cell, we represent the tape state at one point, the state the turing machine head would have if it were here now (whose value doesn’t matter if it’s not here), and one bit saying whether the head is here now. Clearly, each such state at time t only depends on its immediate neighborhood states at time t-1, which is all we need for this to work as a CA.