Brogan

If a minion was defeated but not removed (saved by Brogan), and it is attacked and defeated again later, does the attacker gain the coins again?

Yes.

If the actual heavy minion is removed and Brogan (who counts as a heavy minion) is the only minion remaining, when does the push happen?

If you remove the actual Heavy, and the Brogan is the only minion remaining the push will happen as soon as you are done with minion battle, since Brogan's ability only works in that phase.

At the end of the round, does leveling up or the wave push happen first?

It is specified. Page 8.