1) Yes you would die from the sanity loss. From the rule book (page 9):
Once the haunt begins, if any of your traits (Speed, Might, Sanity, or Knowledge) moves down to the skull symbol, your explorer dies.
This does not differentiate in any way between taking direct damage and any other forms of stat loss, so gaining the Madman can kill you if your sanity is already at its lowest point.
2) No, the Dog cannot carry the Madman (or the Girl). According to the rule book (page 7):
Some omens, like the Dog, aren’t items at all, so they can’t
be dropped or traded. They’re companions that follow
the explorer who has custody of them. (The Dog, Girl, and
Madman all work this way.)
and according to the Dog card:
Once during your turn, the Dog can move to any explored room up to 6 spaces away, using doors and stairs, and then return. It can pick up, carry, and/or drop 1 item before it returns.
The Dog explicitly states that it can only pick up items. The rules explicitly state that both the Madman and the Girl are not items, so therefore neither of them could be carried by the Dog.