The elven king thought the dwarves were there to steal his treasures.
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Yes, in chapter 9 the dwarves (but not Bilbo) were imprisoned by the Wood-elves in Mirkwood forest.
The dwarves were captured and imprisoned by the elves in Mirkwood. Bilbo had to keep entering to form a plan and carry it out.
I believe it was when Gandalf was leaving the group at the edge of Mirkwood.
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).
The dwarves are Bilbo's friends for most of the Hobbit.