Can a simple closed curve in a compact surface be dense? I do not see an argument immediately that it cannot be, but it feels dubious. Does genus have anything to do with it?

A simple closed curve in a surface $X$ is a continuous injection $f:S^1\to X$.  Since $S^1$ is compact, the image of $f$ is compact and hence closed.  So, the image cannot be dense (the image cannot be all of $X$ since $f$ is a homeomorphism to its image).
More generally, the same argument applies to any Hausdorff space $X$ which is not homeomorphic to $S^1$.
