Short answers, pulled from the story.
The Adventures of Tom Bombadil was first published in 1962. Some editions, including the Unwin Paperbacks edition of 1975 and Poems and Stories, incorrectly state 1961, but Tolkien's letters confirm 1962 as the correct year.
The Adventures of Tom Bombadil contains 16 poems. Two feature Tom Bombadil, three also appear in The Lord of the Rings, and only one, "Bombadil Goes Boating", was written specifically for the collection.
W. H. Auden considered "The Sea-Bell", subtitled "Frodos Dreme", to be Tolkien's best poem. The poem draws on medieval dream vision poetry and Irish immram voyage traditions, ending on a note of alienation and disillusion.
The original 1962 edition was illustrated by Pauline Baynes. A later edition was illustrated by Roger Garland.
The book contains background information about Middle-earth found nowhere else in Tolkien's published writing, including the name of the tower at Dol Amroth and the names of the Seven Rivers of Gondor.
The book originated from a request by Tolkien's aunt, Jane Neave, who wanted a small book about Tom Bombadil in the style of Beatrix Potter's Little Books. Tolkien's publisher preferred a larger volume, so Tolkien assembled existing poems he felt Hobbits might enjoy, grouped by theme and accompanied by a mock-scholarly preface.