analytic continuations

To answer your specific questions, a) a maximal (by inclusion) domain does not have to exist. Consider $sqrt{1-z}$ in the unit disk. That is actually the reason why Riemann introduced Riemann surfaces.

But there are maximal domains in various other senses. Suppose you have an analytic germ at infinity. Then there is a unique (!) set $K$ in the plane of
of minimal logarithmic capacity, such that our germ has a (single-valued) analytic continuation to $Cbackslash K$. This is due to H. Stahl.

b) The answer depends on what you exactly mean by a criterion. There are necessary and sufficient conditions for existence of an analytic continuation
(multiple valued) to a given region. Not surprisingly they are difficult to check. See, for example
MR1711032
Atzmon, A., Eremenko, A., Sodin, M.
Spectral inclusion and analytic continuation.
Bull. London Math. Soc. 31 (1999), no. 6, 722–728.

There is a necessary and sufficient condition for a function defined in the unit disk to have an analytic continuation to a given point on the circumference (Euler).

And there are many separate necessary (but not sufficient) and sufficient (but not necessary) conditions in terms of coefficients in various special cases. This is a very vast subject. A good survey of these conditions can be found in Bieberbach, Analytische Fortsetzung, Springer, 1955, and
V. Bernstein, Lecons sur les progres recents de la théorie des séries de Dirichlet, Paris: Gauthier- Villars. XIV, 320 S. (1933).