I've completed all the quests and the went back to some old locations and some enemies were there again so I did my thing and cleared them out again.

When looting I picked up Red Eagles fury sword and wanted to store it at my house, but it won't let me and throws that "Quest Items can't be removed" message.

Is this a bug since I completed this quest a long time ago?
Can I give this item to my follower? Can I equip it and then drop it? Can I sell it or trade it?

I just want to get rid of the weight at this point.

L_2011

What platform are you on? You can remove quest items on PC via the console.

I am on PS3. I've never seen a way to remove a quest item unless it completes a task.
I think this is a bug since it's still listed as a quest item long after this item was used to complete the quest/task

Quest items are weightless.

go back to red eagle's cave and put the sword in the stone and walk out? i always end up double triggering this quest too, it's annoying.

I understand you found the sword twice? Then go to Rebel's Cairn and place the duplicate on the pedestal (or whatever it was), just like you did the first time. You won't get Red Eagle's Bane again, but it should remove the sword from your inventory.

I've had plenty of quest items that had weight and could not be dropped.