Does lapse rate apply below sea level?
Yes.  The height of the ground or ocean surface is of no consequence to the lapse rate as long as we remain above it -- in the atmosphere.  E.g. the dry adiabatic lapse rate is a conservation of entropy as a parcel of air rises or descends.  This means the temperature change is related only to the expansion of contraction of the parcel as it responds to changes in pressure.  When you move down below sea level there is nothing fundementally different than starting at 500 m height and moving down from there.  Both are moving down into higher pressure and the parcel will contract and warm as a result.