No, why would he? He is departing Middle-earth for the Undying Lands in the Far West to be healed, he can never be healed if he stays in Middle-earth.
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Frodo is actually sitting a ways back in the cart from Gandalf, but because the seats they are on are not actually shown and compared, and the side of the cart is the same, the audience gets the effect that Frodo is just smaller, and not actually sitting back.
Most of it is Frodo, with a few parts by Samwise.
They are loyal to Frodo. They are very obsrevant, for example they knew what Frodo was about to say when he was about to say that he was leaving. Also, even though they are afraid, they are still willing to go with Frodo on his journey, even though they knew that they will never come back. * Sorry for using 'they' instead of 'Sam', 'Pippin', or 'Merry' too much.
The hobbits go back to the Shire, and Sam marries Rosie and they have 2 kids. Then the elves (apart from Arwen) leave for the undying lands, and Bilbo, Frodo and Gandalf go with them. Aragorn and Arwen are married and King and Queen of Gondor.
Frodo went with Bilbo, Gandalf, the Ring-Bearers and the elves to the Grey Havens (the Undying Lands), across the sea. There he will be free of the pain he had to endure in Middle Earth as a result of his carrying The Ring. Unfortunately, it was a one-way journey. In other words, he never came back to the Shire.