Why does a polynomial-time language have a polynomial-sized circuit?
Cook-Levin gives one circuit for all inputs of a given size. So although the circuit depends on the input $x$, it only depends on the size of the input. So given TM $M$ with running time $t$, and a number $n$ (the size of input), Cook-Levin gives a circuit $C_n$ of size roughly $t^2$ that solves the problem on all inputs of size $n$. The circuit $C_n$ does not depend on what are the bits of the input for $M$, however we need to know the numbers of input bits as that is going to be the number of input bits for the circuit.