It is possible for Cheyenne to die while fighting the geckos, and if this happens, Sunny's dialogue will change slightly to show her remorse. However, it has no major impact on the quest and Sunny will continue to train the Courier.
Cheyenne will turn hostile if the player kills Sunny.
Cheyenne will attack anyone who attacks the Courier if they are close enough together during the quest to kill the gecko, even if it is Goodsprings residents. It is possible for the Courier to make a Goodsprings settler hostile, then run to Sunny, start the quest, and have Cheyenne kill the attacker.
If Sunny Smiles dies and Cheyenne does not, and the Courier gets far enough away, Cheyenne will sometimes return to Sunny's corpse and will not move.
If Sunny Smiles dies, Cheyenne will also sometimes return to the Prospector Saloon, failing the quest.