1. Characters with negative toughness due to Enhancements are discarded as the first step in battle resolution, so the Hero would not get the LS.
2. There are currently no rules limiting the number of Sites in battle.
3. Side battle cards create a 2nd battle in the Field of Battle, so all cards in both battles remain in the FoB. Ambush the City is probably the only "another battle" card that sets the original battle aside.
4. Numbers on Enhancements do not change the numbers of the Hero. So a Hero decreased to 0 or less toughness through it's own numbers (including other increase/decrease, experience credit, etc.) would be discarded even if there are numbers on Enhancements they are currently using.