In the time of the Roman Republic or Empire did women fight gladiatrix battles?

Yes, but they would not have been called gladiatrices. Gladiatrix is an unattested feminine form invented by modern scholars to make it easier to understand what they're talking about. Latin texts called them either mulieres or simply feminae. They're exceedingly rare but they are menioned several times by our texts--see my post here.

Did they fight animals?

No, not really. at least not what you could call a gladiatrix. Gladiators did not participate in wild beast hunts, which were the specialty of venatores. Cassius Dio mentions some rather unimpressive female venatores but these aren't actually the same thing as gladiators

http://www.jstor.org/stable/3185234?seq=1#page_scan_tab_contents

Female gladiators, or females dressed as gladiators, have been depicted on a marble slab.

Stephen Brunet's "Women with Swords: Female Gladiators in the Roman World" seems like a good resource to check out.