The Witch Hunter can ONLY carry items for you, he cannot use them.
When The Witch Hunter dies, he will "drop" all the items he is carrying, which can then be picked up if the chapter card has finished, or at the end of that round of combat. 
Companions can rest, even if they have full health, however don't forget only one survivor can rest per round, which includes companions!
If the SPECIAL ABILITY side of the die is rolled, unless it's during a round of combat that the enemy will be defeated in, the ability automatically activates.
The TEMPORAL VORTEX curse (from Adventure Pack 1: Cult of The Death Knight), would cause the witch hunter to lose any relics he is carrying.
The CRIPPLING INJURY curse (from Adventure Pack 1: Cult of The Death Knight) would not affect him, as thematically he is carrying the items in his backpack, so losing an arm means nothing to him!

When MARKED FOR DEATH (from Adventure Pack 1: Cult of The Death Knight), if The Witch Hunter falls to 2HP he must use the cult die when rolling. If they were to rise back to 3HP they would no longer need to use the cult die.  

If they pull plague, follow the usual rules for plague as if they were a normal survivor, following the same thresholds.