Weird unsolved shit:
What the fuck is in that passage Beren had to come through to get from Dorthonion to Doriath?
Entwives. Was the tree that Sam's rope magically untied from an entwife? What about the thing the hobbits saw on the north moors? Or Gondor's magic wooden items?
Alatar, Pallando and whatever the fuck happened in Rhun
Why Arnor split into three kingdoms (there's an old myth with a seeing eye, a healing apple and a power item which would fit well)
Gandalf's adventures with Belladonna Took
Grima fucking Eowyn in the eye socket