it will air in? Do producers of the parent show have any input? Or is it Network Executives?

I think somebody flips a coin.

I don't know but I doubt that the actors have anything to say about it. Unless you're Bill Cosby or someone like that. I expect that the cigar filled room full of smartypants execs is where those decisions are made. Often times the slot will change a time or two before settling in. By the time it settles in the show gets yanked off of the air. Many times when a program gets dropped, due to low ratings, the poor judgement by the execs who kept moving the time slot around were partly to blame.