When you discard cards at the beginning of the game, where do they go? The appearance is that they get shuffled back into the deck. However, after discarding key cards and not receiving them the whole game, I am now wondering if they get put in the bottom of the deck. Where do they go after being discarded?
They get shuffled back into the deck before your cards are drawn again. I've had a case where I put back 2 Ironbark Protectors and got back 1 on the draw.
You not drawing them is just unlucky.
Edit: According to Blizzard, that is not the correct functionality of the Mulligan.
The correct functionality of mulligans is as follows: You cannot get back the same card you just mulliganed. You can get back a card you mulliganed as your first draw, and you can mulligan into a second copy of a card you have in your deck, but you cannot get back the same card you just mulliganed back into your opening hand. The card you mulligan are indeed still set aside, new cards are drawn from the deck, and the mulliganed cards are shuffled back into the deck.
If there are instances where mulligans are not functioning as stated above, those can be considered bugs and should be reported on the Bug Report Forums with as much information as you can provide.