The set of embedded minimal surfaces $\Sigma$ with $\textrm{genus}(\Sigma) \leq g_0$ and $\textrm{area}(\Sigma)\leq A_0$ is compact for a generic metric (this follows from White's version of the Choi--Schoen compactness theorem https://mathscinet.ams.org/mathscinet-getitem?mr=880951, see also Appendix A: https://mathscinet.ams.org/mathscinet-getitem?mr=1778099).
However, without the area bounds, this fails: https://mathscinet.ams.org/mathscinet-getitem?mr=1728019 (generic metrics without compactness of stable spheres), https://mathscinet.ams.org/mathscinet-getitem?mr=2026836 (same for higher genus), https://mathscinet.ams.org/mathscinet-getitem?mr=2103999 (positive scalar curvature). In the first two papers, the surfaces constructed are also stable.
Edit: The second part of the question (generic PSC metric) is trickier. I think that no counterexample is known but I would guess that one exists. The tricky part is that for $(M,g)$ a $3$-manifold with a PSC metric, the space of bounded Morse index surfaces is compact. So any counterexample would necessarily have unbounded Morse index (as in the Colding--De Lellis example above). This rules out minimization, which is the tool used to construct the other examples.
One reason to think that such surfaces exist comes from White's degree theory. See (2) in the introduction here https://mathscinet.ams.org/mathscinet-getitem?mr=978083 (obviously not exactly your question since it deals with disks not closed surfaces).