It should be pointed out that your definition of finite type is not the usual one. The usual definition is that a Riemann surface is of finite type if it is conformally equivalent to a compact Riemann surface minus a finite set of points. For instance, under this usual definition, an annulus of finite modulus is not of finite type.
Donu Arapura is correct that the only exceptions to hyperbolicity are the usual ones, which answers 2.
As for 1, such things are classified topologically by the space of ends, by a theorem of I. Richards, see this. When you use the usual definition of finite type, then there is a Teichmuller theory for non-finite type surfaces with finitely generated fundamental group, where you specify boundary values when solving the Beltrami equation. You may read about this is most analytic treatments of Teichmuller theory, such as the books by Gardiner, Gardiner-Lakic, Nag, et cetera. I don't know about the theory in the infinitely generated case.
As for 3, any countable group is the automorphism group of some hyperbolic surface. The idea is that groups arise as automorphism groups of their Cayley graphs, and fattening the Cayley graphs into surfaces provides the result as long as you chose the lengths of the meridians carefully. This is a theorem of Allcock, see this.