1. If the hero is the only survivor, but hasn't got the stone yet, and the boss is still alive, does he/she need to go on playing solo till he/she defeats the boss and runs out with the stone?
Heroes aren't killed (only knocked down) until the Boss is defeated and endgame begins, so the scenario in your question can't happen. However, after endgame has begun, if only one Hero remains, that player is the winner and doesn't have to get to the Entrance tile.
2. What is the difference between "kill a monster" and "defeat a monster"?