TTD gradually and severely lowers the damage resistance of the target goaded by TTD to all damage sources. DoTs and direct damage are both affected by TTD's debuff, which becomes more severe as time passes.
It only affects the single target goaded by TTD, no other enemy is hit by the effect. Other targets goaded via other abilities like war cry and challenge while TTD is active on another target does not apply the TTD debuff to those other enemies.
As for the last part, TTD increases the damage output of the target goaded by TTD proportionally to the scale of their damage resistance debuff - increasing as time goes by. It is a buff applied to the enemy, and not a debuff applied to the player. If enemy B is the target hit by TTD, enemy A's fire DoT will not do increased damage to you.
However, if enemy B hits you with a fire attack, their DoT will be super strong, but only after enough time has passed for TTD to buff their damage output to sufficient levels. Most enemies will be dead by the time this becomes noticeable though, and the DoT will most likely be on the champ of the group who used TTD to goad a foe anyways, who could probably shrug off pretty much anything anyways.