The battle does not immediately end, but a character in battle that changes alignment (Hero to EC, or EC to Hero) returns to territory as part of the convert. There have been cards printed in recent sets that would allow you to add a hero or EC after one has been removed by a convert or another ability.
Heroes can fight heroes, but that's a different ability called side battle.
If a character's toughness is reduced to 0, they are discarded, regardless of why it's reduced to 0.
