At the end of the book, why is Bilbo no longer considered to be respectable by the other hobbits? What special abilities do hobbits, in general, have that makes Bilbo useful to the dwarves?

In The Hobbit by J. R. R. Tolkien, Gandalf explains why he recommends that Bilbo accompany Thorin's band of dwarves as the burglar:

If I say he is a burglar, a burglar he is, or will be when the time comes. There is a lot more in him than you guess, and a deal more than he has any idea of himself.

In a general sense, Tolkien writes that any hobbit is "as fierce as a dragon in a pinch," and that Bilbo's great grand-uncle Bullroarer was so brave that he charged the ranks of goblins in a battle and lopped off their king's head. He also explains that hobbits can disappear quietly and quickly when larger folks are about.
Tolkien explains further about the special abilities of Hobbits that would make Bilbo useful to the dwarves in the section of the prologue to The Fellowship of the Rings called "Concerning Hobbits." He writes that hobbits are quick of hearing, sharp-eyed, and nimble and deft when they move around:

They possessed from the first the art of disappearing swiftly and silently when large folk whom they do not wish to meet come blundering by; and this art they have developed until to men it may seem magical.

At the end of his journey, Bilbo was no longer considered respectable to other hobbits because hobbits do not like adventure. They look with suspicion on anyone who finds it necessary to travel far from their homes. Tolkien makes this clear in the third paragraph of the first chapter of The Hobbit when he writes:

The Bagginses had lived in the neighborhood of The Hill for time out of mind, and people considered them very respectable, not only because most of them were rich, but also because they never had any adventures or did anything unexpected.