"If a player loses a challenge as the attacker, that player must fulfill the claim of that challenge, using the Claim value on the defending player's plot card"

If I'm playing a four player melee, and I put Doran Martell into play, does this effect apply to everyone - even when they're attacking each other? How about if I was the losing attacker?

It simply says "a player", so it applies to any player attacking another player, you included.