This is the suffix ‑s, of which the paywalled OED says:
A shortened form of the hypocoristic diminutive suffix ‑sy suffix², added to the same classes of words, as Babs, Toots; ducks (see duck n.¹ 3c), moms.
Moreover, they point out that its voicing follows the same rules used when making plurals or possessives.
This suffix does not affect stress and does not add a syllable. It is pronounced as a terminating consonant on the preceding element and is consistent with the voicing of the immediately preceding sound, hence e.g. ducks Brit. /dʌks/, U.S. /dəks/ but moms Brit. /mɒmz/, U.S. /mɑmz/.
As for the referenced ‑sy suffix, this one they explain in more detail:
Hypocoristic diminutive suffix added to (i) proper names, as Betsy,
Patsy, Topsy, also in the form ‑cy, as Nancy, (ii) common nouns, as
babsy, ducksy, mopsy n., petsy, popsy n. (popsy-wopsy). In adjectival
formations expressing a degree of mocking contempt, as artsy-and-craftsy,
artsy-fartsy, backwoodsy, bitsy, booksy, folksy, itsy-bitsy, teensy,
etc., the suffix may be considered to represent a nursery form (cf. ‑y
suffix⁶), or the plural (or even a singular ending) in ‑s + ‑y suffix¹.
And for its pronunciation, they note:
Primary stress is retained by the usual stressed syllable of the preceding element and vowels will be reduced accordingly.
As for your second question, there are a few male uses like Chubs or Pops, or as the OED mentions, popsy-wopsy, which is bit mocking. But those don’t start out as a longer name and get cut down.
For male nicknames that actually appear to use this suffix because it isn’t just a shortening that cuts off at just the right place like Tristram > Tris, Dexter > Dex, Basil > Baz, Alexander > Lex, Caspar > Cass all do, you do have cases like Charles > Chaz, Quentin > Quince,
Reginald > Rex,
Robert > Bobsy,
Laurence > Lars,
Julian > Jules,
Patrick > Patsy.
But there really aren’t at all so many as there are the female shortenings such as you mention.