Can I make an item in an NPC's inventory un-removable? If I make it a quest item it doesn't work and if I transfer it I can't drop it.

What exactly do you mean un-removable? Just that the player can't remove it when they are dead? If its armor or clothing you can remove the check mark "playable". If it's something else put "removeitem TheItem 1" in an ondeath block.

It's for a companion. I want it so that the player cannot remove the item from their inventory.

Well something like this will work then since I'm guessing its not an item that can be flagged as not playable...Put this in the companions script.

short removeBegin MenuMode 1008if remove == 1   player.removeitem MyItem 1   additem MyItem 1   set remove to 0   message "you can not remove this item";remove this if you don't want the messageendif if player.getitemcount MyItem > 0   set remove to 1endifEnd

begin onadd playerRemoveme TheCompanionRefend

Is probably a bit easier.

How about this simple idea that doesn't require a script.
So go to the clothing item you want.
right click edit, change the ID (e.g. from MiddleClassPants01 to MiddleClasspants01Companion)

then uncheck the 'playable' box.
everything else doesn't matter for this.
then drag'MiddleClassPants01Companion' into the NPC's inventory rather than MiddleClassPants01.
Done.
Hope that helps

You are the master of not reading a whole thread before posting I see.