Yes, you can claim anything about the individual members of the empty set and you are always correct (saying "there are 3 members of the empty set" doesn't count because you are talking about the collected members, not the individual members). Saying "All X are Y" is the same as "No X are not Y" or "there are no X that are not Y" which is always true if there are no X, period.
But I think that "monkey" is common English usage for "primate". "Monkey" has a more precise meaning in biology, a fact that I completely ignore unless I'm dealing with on-duty biologists. (:
(but my friends call me "Tye")