I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward.

Categorizing by genus: for $S^2$ ($g = 0$) this is evidently true, and for $T^2$ ($g = 1$), one can take the radius of the tube to be very small relative to the outer radius (see e.g. here) to force mean convexity. I have no idea if this is true for higher genus.