Did Bilbo Baggins die

Frodo left the Shire and went to the Undying Lands where he lived with the elves and doesn't die.

Frodo sailed to the Undying Lands to live out the rest of his days in the pure beauty of Valinor. It was not within the authority of the Valar to grant immortality to any being not created so. Tuor was the sole exception to this role.

No, they could not give him more life. He went to Valinor to be taken care of and to find rest. There was a story called Frodo's Dreme(sp) where he was mostly lonely in Valinor and died unhappy. It was a dream he had before he went and wrote it in the Red book of Westmarch.