In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic heptagons".

Question 1: What does this tiling look like?

Question 2: Is it always possible to tile a genus $n$ surface by $f$ regular $n$-gons with interior angle $\pi/v$ (so that $v$ faces meet at every vertex) as long as the restriction given by the Euler characteristic $$\chi=2-2n= f-nf/2+nf/v$$ is satisfied? Answering this question, Igor Rivin says yes, but it seems his argument only shows that regular hyperbolic $n$-gons of interior angle smaller than the interior angle of a euclidean regular $n$-gon exist.