If someone dies while killing the boss (i.e lands the final hit but is killed in the process), are they out of the game completely? Or are they deemed to have died before the end game started and thus simply knocked out?
The endgame doesn't start until someone picks up the Summoning Stone, so that player is just knocked out.
Secondly, say someone picks up the summoning stone on their turn, do they immediately get the first player token, and hence get another turn immediately after? This was a friend's interpretation of the rules, but it seemed incorrect to me because it basically let him get enough of a head start to run straight to the entrance for the win.
You are correct. The rules say the hero equipped with the Summoning Stone takes and keeps the First Player token at the beginning of each round. A new round doesn't begin until play gets back to the current First Player and monsters roam.