If the creature or planeswalker with a bounty counter on it ceases to be a creature or planeswalker, Chevill’s last ability will still trigger if that permanent is put into a graveyard from the battlefield.

Chevill’s middle ability and last ability look for any bounty counter on your opponents’ permanents, not just one from that Chevill. For example, if players A and B each control a Chevill and player A has put a bounty counter on player C’s creature, player B’s Chevill won’t put a counter on anything and both players will gain 3 life and draw a card when that creature dies.

If your life total is brought to 0 or less at the same time that a creature with a bounty counter on it is dealt lethal damage, you lose the game before the last ability goes on the stack.

If Chevill leaves the battlefield at the same time that an opponent’s permanent with a bounty counter on it dies, you gain 3 life and draw a card. If Chevill leaves the battlefield before that permanent dies, Chevill’s last ability won’t be around to trigger.