Tom is great; he's one of the best of Tolkien's characters, I think. If I might be allowed a small correction - the Dutch doll called Tom Bombadil didn't belong to Priscilla; nor did she ask JRR to write a poem about him. The name comes from one of the stories which Tolkien told to his children, but did'nt write down, some time in the 20s, before Prisca. was born.
The doll belonged to Michael. John apparently took a dislike to the doll, and pushed him down the lavatory. Thankfully, he was rescued without too much damage.
The term 'Dutch doll' refers to the style of the doll; having jointed wooden arms and legs.
As for Tom - he made his first appearance in the Oxford Magazine in 1934, well before LotR was published. Tolkien was fond of Tom - he published a book of poems in 1962 called 'The Adventures of Tom Bombadil', which includes two poems about Tom, and a nice piece in the introduction about his relationships with the hobbits of Buckland - Tolkien says his name is Bucklandish in origin.
I like Tom, and his singing. But, as Ursula le Guin noted in a paper, Tom's whole mode of speech is rhythmic; meant to be read aloud. The best way to appreciate this is to hear it read by Rob Inglis on the LotR tapes. Bombadil is ancient; he doesn't conform to anything else we know. And he is Master - when he says 'Whoa!' to Frodo and Sam, they stop suddenly as if struck. When B. says 'Show me the precious ring', Frodo does so instantly. Look out for that next time you read; everyone and everything does as he says; but no-where do we get the idea that he's forcing anyone - he is Master, and in his realm everything does as he says.
Except the lintips. These little creatures appear in a third Bombadil poem; only ever printed in two obscure books during the 60s, now long out of print. Bombadil seems quite taken with them.
Old Tom Bombadil is a merry fellow. Tolkien liked him; that's the reason he's in so many of Tolkien's books. But Tolkien isn't giving very much away about Tom - he once wrote that Tom is an enigma - intentionally.