The categorical answer may depend on any scheme you are under or US requirements. My comments below are what my site allows for it's staff (we are UK based with BRC).
Employees are not allowed hair and beard nets when they step outside; can they store the hair and beard nets in their pockets until they come back inside?
Hairnets and snoods should be disposable; my staff are expected to dispose of these on exit and put on fresh ones when re-entering. The PPE can become contaminated outside and the inside of one's pocket is not exactly a controlled environment or one that can be properly maintained and tested.
if an employee only has a mustache, does a face masc that covers that area count as a beard net?
I require any member of staff with a beard, moustache or "excessive stubble" to wear a snood. If you can get something that covers the moustache only as Glenn says then I expect this could be ok, though I think a good old fashioned snood is a safer option.
Previously during inductions and training I've received similar questions, usually "but I'm bald, do I still need to wear a hairnet?". My answer is yes - because it avoids the situation of having the blurry line of assessing baldness/hair length, makes life easier for everyone and ensures compliance.