In my opinion,
Liv is maybe OK as an occasional nickname for
Olivia, but DEFINITELY NOT for a stand-alone name. It is too monosyllabic for a stand-alone name in my opinion.
Liv Ariadne and
Liv Theodosia are no-goes for me, unless
Liv's real name is
Olivia. I love all the names
Olivia,
Ariadne and
Theodosia though. Maybe
Liv as a middle name, but not my first choice. (i.e
Ariadne Liv or
Theodosia Liv).