Yes, you can derive the Klein-Gordon, (free) Dirac, (free) Maxwell, linearized Einstein vacuum, etc., equations from the representation theory of the Poincaré group.
Yes, you can derive the ordinary non-relativistic (free) Schrödinger equation from the representation theory of the Galilei group (i.e., the represenation theory gives you the well known explicit form of the free non-relativistic hamiltonian).
The definitive and ultimate reference (assuming you want rigorous mathematics, which for this type of topics is the only way I recommend) for this topic is: "Geometry of Quantum Theory", by V.S.Varadarajan: http://www.amazon.com/Geometry-Quantum-Veeravalli-Seshadri-Varadarajan/dp/0387493859
I warn you, though, it's very hardcore material, not for the faint of heart.
Complementary references are: Folland's "Quantum field theory, a tourist guide for mathematicians" and Folland's "A Course in Abstract Harmonic Analysis".
Also, Theory of group representations and applications - A.O.Barut, R. Raczka.
Valter Moretti's book (he posts in this site) "Spectral Theory and Quantum Mechanics" also has useful material on quantum symmetries (Wigner's and Kadison's symmetries and theorems, Bargmann's theorem, stuff on central group extensions and also some stuff on the Galilei group).
Some online articles and notes: http://arxiv.org/abs/0809.4942v1; http://www.staff.science.uu.nl/~ban00101/lecnotes/repq.pdf