Brogan

What is the significance of referring to Brogan as a heavy minion? Does it mean he is immune as well?

Correction: 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.