This is only a partial answer.
First, a sphere does not admit Fisk triangulations. Fisk has proved it by using the 4-color theorem, but there is a simpler proof using Fisk's method of coloring monodromy. Try to color the vertices in three colors. There will be no contradiction when you go around an even vertex; but the colors will be permuted when you go around an odd vertex. For one vertex this will be the $(12)$ transposition, for the other vertex $(23)$ transposition. Thus, the fundamental group of the sphere minus two points will be mapped epimorphically onto $S_3$, which is a contradiction.
The torus does not allow triangulations with vertex degrees $5, 7, 6, \ldots, 6$ (even if you don't require the $5$- and the $7$-vertex to be neighbors). This is a result by Jendrol and Jukovic, reproved recently by a geometric method. There is however a "triangulation" (which is not a simplicial complex) with two vertices, one of degree $3$, the other of degree $9$, and I strongly believe there is a simplicial complex with this property.
For the surfaces of higher genus I would guess that one can have all vertex degrees $6$ except two odd neighbors. This guess is based only on the fact that the combinatorial (coloring monodromy) and the geometric (holonomy of a Euclidean cone-metric) obstructions don't forbid this.