The oxymercuration is stereoselective and stereospecific.

The demercuration process is not, since it involves a free radical with an indefinite lifetime that can potentially invert and give rise to mixtures of syn and anti addition.

It would lead to a racemic mixture of enantiomers (assuming R is achiral).

If you’re talking about selectivity for different amines (primary vs secondary) it would still be “regioselectivity” you’re talking about. If it were selectivity for an amine versus an alcohol (i.e. different functional groups) then that would be “chemoselectivity”.

“A reaction is termed stereospecific if starting materials differing only in their configuration are converted into stereoisomeric products. According to this definition, a stereospecific process is necessarily stereoselective but not all stereoselective processes are stereospecific. ” That’s from IUPAC, here. https://goldbook.iupac.org/terms/view/S05994 .

That’s slightly different from the way the term is commonly used, as I believe it is in your case. Let me explain.
Epoxidation ALWAYS gives the “syn” epoxide, and NEVER the “trans”. It’s a consequence of the mechanism.
Take cis-2-butene and trans-2-butene and epoxidize them. cis-2-butene will give you cis-2-butene epoxide, and trans-2-butene will give you trans-2-butene epoxide, and there is no leakage between the two. That’s generally what’s meant by SPECIFIC.
In Sharpless epoxidation, you’re starting with an achiral alkene (actually allylic alcohol). After the Sharpless you might end up with a 99:1 mixture of enantiomers. Although 99:1 is very high, it’s not “specific”, it’s “selective”.

Like I said that’s the common usage, which is a little different from IUPAC’s sometimes.

A good suggestion. Thank you.