What are the consequences of a faster algorithm for $CIRCUIT$-$SAT$?
I'm not sure how to answer (1), but (2) is known to imply circuit lower bounds against NQP (non-deterministic quasi-polynomial time). This is from Cody Murray and Ryan Williams' STOC 2018 paper.

In fact, they show that these lower bounds follow from faster algorithms for what they call Gap Circuit Unsatisfiability: given a circuit $C$ on $n$ variables and the promise that either (a) $C$ is unsatisfiable, or (b) $C$ has at least $2^n/4$ satisfying assignments, determine which of (a) and (b) is the case. This problem is easily solved with randomness, so this hypothesis is perhaps more likely than a faster algorithm for Circuit Satisfiability. 

Ryan has a very nice talk on this thread of work; a recording in two parts is available here and here.