At 10.3, we aren't doing any vertical transformations. Some support was added to 10.4.1 and Pro 1.4, with more coming in Pro 2.0.
For your question, "I believe ArcMap is performing both a transformation to a different horizontal datum and a projection from a flat grid based state plane system to an elliptical geographic system", this isn't quite true.
The data is being converted from a projected coordinate system, plus a gravity-related vertical coordinate system. The target is a geographic coordinate system and ellipsoidal heights based on the geographic coordinate system. At 10.3, what you would actually get is WGS 1984 for the geographic coordinate system, and the vertical/z values are still referenced to NAVD88. Vertical coordinate systems are...complicated. Most are only loosely tied to a horizontal (geographic or projected coordinate system) and even though usually because there's a transformation method that uses a grid/file of offsets and you have to have gcs connected to it to interpolate the offsets.
In this case, at 10.5, we would convert the NAVD88 values to nad83 ellipsoid heights using the Geoid12b geoid model, then convert from NAD83 + NAD83 ellipsoidal heights to WGS84 + WGS84 ellipsoidal heights. If the target VCS was WGS84 geoidal heights AKA EGM2008 heights, the third step would be to convert the WGS84 ellipsoidal heights to geoidal using the EGM2008 geoid model.
Melita