If, at this point of the Prof's development of Middle Earth, the eagle's are indeed the messengers of Manwe, then they would leave as soon as their job was done.  They have come to help out at the end, as they always do, after the people of ME do what they must and what they can.  As to the word 'chief', I am no scholar, but perhaps their community is loosley knit, and they do not have a 'king', though I thought Thorondor was refeered to as King of the Eagles, but I dont have time to look at the moment.  And the chief is a bird, so I am sure he likes shiny objects and would not turn down a crown of gold :)

As for Bilbo's wish, I think that it is because of the fact that his Adventure has turned into nothing more than a war between states (races in this case, I guess), and wishes to be back in his previous adventures that were not as mundane, or tied to the everyday world.  Wars are a constant in human life, but flying with the eagles are not, nor riding down a river after escaping from the elves, or talking to a live dragon!  Much like Frodo, he is now 'waking from his dream' and wishes to return to it.

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Then Morgoth hurled aloft Grond, the Hammer of the Underworld, and swung it down like a bolt of thunder.  But Fingolfin sprang aside, and Grond rent a mighty pit in the earth, whence smoke and fire darted.  Many times Morgoth essayed to smite him, and each time Fingolfin leaped away, as a lightning shoots from under a dark cloud; and he wounded Morgoth with seven wounds, and seven times Morgoth gave a cry of anguish, whereat th hosts of Angband fell upon their faces in dismay, and cries echoed in the Northlands.

                       -- Of the Ruin of Beleriand and the Fall of Fingolfin
                          The Silmarillion