The answer to your first question is No, they are not all homeomorphic. In the first question you did not insist that the a<sub>n</sub> converge to 0, and so let us entertain the idea of other crazy sequences. For example, we might let a<sub>n</sub> enumerate all the rational numbers. In this case, we would have circles of every rational radius. This is clearly not homeomorphic to the ordinary Hawaiian earring. For example, every convergent sequence in the ordinary Hawaiian earing lays on a path, but this is not true for the crazy dense version, since every point will be a limit of points on other circles.

You can make a less-crazy counterexample by having just two limit points in the sequence a<sub>n</sub>. For example, let a<sub>2n</sub> converge to 1/2 and a<sub>2n+1</sub> converge to 0. This example would be compact, but still different from the classical earring.

A similar arguent shows that any two sequences with different finite numbers of limit points will be non-homeomorphic. I believe that the homeomorphism type of the resulting earring will be determined by the homeomorphism type of the set {a<sub>n</sub>}, plus the question of whether 0 [Edit: and infinity] are limit points.