Does vertical datum get projected? I feel a bit confused with the concept of the vertical datum and doing a projection.
No, when doing a map projection, the vertical coordinates are unaffected. If you look at the math behind a map projection (cf. Snyder's Map Projections: A Working Manual). you'll see that there's no mention of any vertical data. The only time you might see altitude or elevation mentioned is if the projection allows you to set its perspective point like vertical near-side perspective. Another projection used in Colombia takes the average local elevation into account to minimize the distortions. That doesn't affect the point elevations in the area though.
You can have 'horizontal' data in latitude-longitude, referenced to a geodetic/geographic coordinate reference system (CRS), or in planar coordinates (x-y or easting-northing), referenced to a projected CRS. The vertical coordinates could be gravity-related like orthometric or normal heights referenced to a geoid surface or leveling surface or ellipsoidal heights referenced to the underlying ellipsoid used by the GeoCRS.
A vertical CRS (AKA vertical datum) identifies what origin point or surface the 'z' coordinates are referenced to.
You can even perform horizontal datum transformations, like from NAD 1983 (1986) and NAD 1927, while still having the same gravity-related heights. Those heights are just transferred unchanged between the two 'horizontal' systems.
If the data has ellipsoidal heights, some types of horizontal datum transformations (Bursa-Wolf, Coordinate Frame, Position-Vector, Molodensky-Badekas, etc.) can convert the ellipsoidal heights between two GeoCRS.
Most datum transformations that use on-disk files (NADCON, NTv2, etc.) only convert the horizontal coordinates, but I've seen an overloaded NTv2 file that had vertical offset information in it as well. Some file-based methods do support both horizontal and vertical offsets and could perform both horizontal and vertical transformations.