Will M. Baker
Resolve all of them. Also remember that these effects are after you resolved the battle. So you would gain 2 diseases and destroy a hero after you won or lost the battle.
Pre-Towers of Ruin, Battle effects are resolved immediately unless the effect destroys a card. So in the above example, you would gain 2 diseases immediately (before resolving the battle), and then destroy 1 hero after resolving the battle.
Not sure there is a card where this matters, but the active player may decide the order in which to resolve the Battle effects if the monster has multiple Battle effects that would trigger at the same time.