Searing Truth

Yes. If your opponent casts this and targets your Warded avatar, you will not draw any cards and the Ward will break. Normal targeting rules apply (e.g. a spellcaster in the void or subsurface couldn't target an avatar with this magic, since they aren't in the same region).

Damage prevention effects on the avatar will still apply to the damage from Searing Truth.

Unless resolving on the storyline, X cost spells are treated as costing zero.