No they are not. Neverwittlesbane (yeah I know I spelt it wrong) to cut the tree, weighs 10 pounds, so no, quest items are not weightless.
well it stops being a quest item when youve finished the quest (unlike the bards college items!!).
Anyway, I left the sword in the 'lock', so this hasnt been an issue. did you take the sword back once youd opened the door?
Wait, youve been to Rebels Cairn to fight the Red Eagle boss, right?