Yes, HYCOM is a full Ocean General Circulation Model (OGCM) and includes wind stress forcing and Coriolis. It does not include Stokes drift.
The wind stress used is in the "ice" data set, as is the near-surface ocean velocity. This was calculated from NAVGEM 10m winds (and its other atmospheric fields) using a bulk parameterrization and HYCOM's surface fields - it is based on U10-Uocn (relative winds).
Alan.