regular tiling of a surface of genus 2 by heptagons
A picture is in the comments to John Baez' Blog Post. EDIT For the second part of the question, no there is no obstruction, as long as the implied angle ($2\pi/v$) is smaller than the angle of a Euclidean regular $n$-gon. This is true by an obvious argument I had originally heard from Bill Thurston: a very small hyperbolic regular $n$-gon has angles close to Euclidean, a very large regular hyperbolic $n$-gon has angles $0$ so, by continuity, somewhere in between you will have the angle you want.
Addressing the second question positively, this is the Main Theorem of this paper:
Edmonds, Allan L.; Ewing, John H.; Kulkarni, Ravi S., Regular tessellations of surfaces and (p,q,2)-triangle groups, Ann. Math. (2) 116, 113-132 (1982). ZBL0497.57001. MR0662119
In fact, they show this also works for non-orientable surfaces, except for the non-existence of a degree 3 triangulation of the projective plane.