Is what the cousin did, in fabricating a fictional character for a young man and concocting a story of his death, considered lying and deceit, pardonable? And is it obligatory to maintain ties with her after her lies were exposed and she persisted in denial, despite the severe psychological harm she caused to the sister and the family?

Your cousin's lying is impermissible, and she must repent from it. Shunning her is permissible if it is expected to lead to a benefit, such as her abandoning the sin. Nevertheless, forgiveness and overlooking are more virtuous, for Allah Almighty says: "So whoever forgives and reconciles, his reward is due from Allah." You must warn your sister against speaking with men online, as it is a path to corruption. Her illusory love indicates her impulsive emotions, which those who seek corruption exploit. Closing this door is the safest path.