If your lone character is about to be discarded, you have initiative (you are losing the battle). If you have multiple characters in battle and one is about to be discarded, you only gain initiative if you are left in a stalemate, mutual destruction, or losing condition after the targeted character is removed from battle.
When a special ability reduces a character to (*/0) or less during the Battle Phase, the character is not immediately discarded. Characters can be saved during the Battle Phase (e.g., setting the character aside, banding the character into battle, etc.). During battle resolution, all characters in play or set-aside areas with ability of (*/0) or less at that time are discarded (see Losing by Numbers in the rulebook [p. 20]).