Does the discard get shuffled into the deck when adding the heat up card?
Yes, it specifies in the rule book that the shuffling of the boss deck means you have to relearn the Boss's pattern. If you only shuffled the deck (not discard) it would be trivial to wait until the entire deck was in the discard, then do the final damage to "heat up" maintaining exactly the same order for the rest of the deck, with the heat up card being next (and therefore last).