Anyone know how to move quest items from the stash to a character's inventory. I can move items from all the other sections (consumables, armor, weapons etc) but the quest items are immovable?

 

I can't use a key in my stash which I need to open a door.

There's an option 'Don't Restrict Stash'. If your stash is restricted, you can only access it in towns, inns, etc. 

 

If you're on Expert Mode, then stash is always restricted. 

 

When it's not restricted, just open stash and drag & drop.

You don't need to move your quest items into your inventory. In fact, I don't think you can anyway. If the quest item is in the stash it should automatically be used where you need it. If it's not working then you either don't have the proper key, or you've encountered a bug.

Anyone know how to move quest items from the stash to a character's inventory. I can move items from all the other sections (consumables, armor, weapons etc) but the quest items are immovable?

 

I can't use a key in my stash which I need to open a door.

You don't have to. If the key is in your stash the door will unlock when you interact with it.

Quest items are automatically placed into stash and cannot be moved. If you have the item inyour stash, it is automatically used in the quest. Maybe the key you have is not for the door you're trying to open.

Quest Items and Enchanting Supplies are always added straight to the Stash and can't be moved from the stash, afaik.