No they don't need to be immortal, but not falling over to a stiff breeze would be nice. Even a fairly basic unit like judicators alone has enough firepower to kill a skink priest potentially in a single turn. Get 2 or 3 of em and it's basicly a guarantee. Look-out-sir rule or not. Hell they don't even need their special bow for it, their regular bows are enough. And it's not like they're unique in this. Similarly even basic spells easily leave most minor heroes like this mortally. Get in 2 or 3 and they basicly guarantee a kill. And these are the only type of units that are killed this easily. Even fodder like skinks usually outlasts that unless you get particularly lucky when attacking them.
As for endless spells, those things generally get to move 8". You need to cover quite a large area to keep them from hitting a hero, especially the ones with smaller models...
Not really, as again, they fall over to a stiff breeze. Most of these support heroes cannot take more than a handfull of attacks. So unless you manage to limit it to only 1 model in range even minor units quickly become a threat. Which is going to be difficult as most support heroes are stuck with a 1" melee profile so it's not like they can at least attack from the relative safety of your backline.