I think that Tom Bombadil is included (in LoTR at least) to provide some light relief and a short rest for Frodo et al, not to mention the reader. This would explain why he disappears, apart from a necessary mention at the Council of Elrond, after they leave the Barrow Downs.
As such, there's not really much need to explain a great deal about him; in fact the enigma makes him more compelling than he would be if we knew the whole story. I do wish he'd associated a name with that brooch, though...
My question would be: Do we really want to pin Bombadil down? Surely he's more interesting as an enigma and there is a saying that we lessen what we define.