As Artemis protected a mother in childbirth, and a girl child until marriage, preforming a miracle heal - so could she suddenly slay a girl child or those who earned her wrath.
So, yes.
You can not kill Artemis, she is a ancient immortal Greek goddess.
Yes, Artemis would kill animals because she is the goddess of the Hunt. It's what she does.
Apollo, in one account of the myth of Orion and Artemis, apparently used Artemis to kill Orion unknowing.
There is no known myth about Artemis killing a cyclops.
yes, however, Artemis was trying to kill them
Artemis aided Apollo in the slaying of Python.
Both, Artemis could kill or heal (with her arrows) women, as Apollo could (with arrows) kill or heal men.
Yes, Artemis was a great huntress and could hunt and kill/catch anything
No, because Artemis was a goddess and that means she's immortal. Which means she can't die.
Most likely Artemis killed her with her arrows. Artemis killed Chione because she became too vain after an affair with Apollo.
Yes, she did, a couple of times in fact.
Artemis didn't kill Orion; most stories say that Apollo did because he was defending his sister's maidenhood after she started getting fond of Orion.