You choose the target of the chaos ability as you put that ability on the stack. When the ability resolves, you choose a value for X and decide whether to pay {X}. If you do decide to pay {X}, it’s too late for any player to respond since the ability is already in the midst of resolving.

When you count the number of charge counters on Kilnspire District, count all charge counters on it, not just ones you put there.