For example, if Brash Taunter is dealt 3 damage, its middle ability deals 3 damage, not 1, to the target opponent. If your life total is brought to 0 or less at the same time that Brash Taunter is dealt damage, you lose the game before its middle ability goes on the stack.

Can Brash taunter fight your own creatures?

PSA: Brash Taunter can target your own creature and still deal damage to your opponent.

Can you target your own creature?

702.11c "Hexproof" on a player means "You can't be the target of spells or abilities your opponents control." 702.11d Multiple instances of hexproof on the same permanent or player are redundant. Which makes it perfectly legal to target your own creatures.