Question about Circuit-SAT NP-Complete Proof
You're right.  All that matters is that the reduction runs in polynomial time.  (It follows that the circuit has polynomial size, since the output of any polynomial-time algorithm must be polynomial in length.)
Given a circuit with multiple inputs, you can force some of the inputs to 0 or 1.  If the circuit has polynomial size, this modification takes polynomial time, since you're only modifying at most polynomially much stuff.