Gandalf believes in Bilbo, and he says in the book that he wouldn't of brought-en him if he wasn't in good use.
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The hobbit Bilbo Baggins, mainly, although he had an awful lot of help along the way.
Bilbo Baggins' mother was called Belladonna Took.
"Bilbo Baggins" is the name of Bilbo Baggins. Unless you're asking for his untranslated Westron name, which would be "Bilba Labingi".
Bilbo Baggins' home is named Bag End.
Frodo knows Gandalf because Gandalf was like Bilbo (Frodo's grand-uncle) guardian on Bilbo's adventures. Now, Frodo knows Gandalf through Bilbo.