Bilbo uses his magic ring and his wits. He follows Gollum with the ring on and is able to leap over him to get to the back door. There he squeezes through, though he lose his buttons doing so.
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Bilbo hung on to Dori's legs when the eagle plucked Dori from the tree. Therefore, both the eagle and Dori were instrumental in helping Bilbo escape from the tree.
{| |- | Dori was the one who helped him. He climbed back down out of the tree and boosted Bilbo up. He was almost caught by the wolves as he scrambled back up into safety. |}
The eagles came and rescued the party from the trees. They were flown off into the mountains and later left not too far from Beorn's home.
Bilbo's escape plan is mediocre. He risked the lives of the dwarves while trying to carry it out.
Yes, in chapter 9 the dwarves (but not Bilbo) were imprisoned by the Wood-elves in Mirkwood forest.
They were imprisoned at the time he revealed the ring to them. Considering Bilbo used the ring to to go invisible and help them escape, it is likely that the Dwarves were grateful. Although they clearly disliked the actual method of escape (hiding in barrels which were tossed in a freezing cold river).
No, the goblins did not get Gandalf. Bilbo's yell when he discovered the goblins stealing things was enough to prevent him from being captured. Gandalf killed a number of the goblins with a big flash of lightening.
The elven frailty is wine. The elvenking's butler and the chief guard drink some wine, which caused them to fall asleep. This gave Bilbo the chance of stealing the keys to the jail cells, and escaping with the rest of the dwarves.