Islamic Rules say that all the livelihood expenses of the wife ( including housing and other usual expenses) is the responsibility of her husband even if she is a rich person. If your wife accepted to pay her own rent not knowing that it is the responsibility of her husband, you must tell her the truth and pay for her rent. If she wants after knowing her right of housing on you, to voluntarily pay her rent, that will be favor from her on you, but you are not allowed to force her or pressurise on her to do so.
No, it is not the shar'i responsibility of a male relative to take care of a single female relative after she gets divorced.
If the relatives wish to do this out of good will, and for the sake of God, then that is a good thing and they will be rewarded for it. Of course, in the case of a woman getting divorced and her not having any immediate family, it is important that her kin support and take care of her and not leave her alone.
With prayers for your success.
You should not abandon hope for her. Continue to encourage her, while not being forceful or pushy. Pray where she sees you and perhaps she will feel remorse that she is not praying.
A scholar in Najaf told me about being persistent, "I remember there was one of the youth who followed my father in taqlid and at some point he became a communist and left Islam. My father would pass by this youths shop on his way to salat everyday. My father would always say salam to the youth, but the young man would ignore him and never reply. My father kept saying salam to him everyday for one year then the youth finally replied to his salam and ended up repenting and coming back to Islam."
Unmarried daughter should look after her old parents who need her with them. This is a religious and moral responsibility on every son and daughter towards their parents. If the parents are in need for their livelihood expenses, it becomes obligatory on the sons and daughters to financially support their parents and fulfill all their needs.
If the parents feel sad or hurt because of their daughter leaves living with them, it becomes her responsibility to live with them and avoid hurting their feelings.
Hadeeth says: Allah created the souls of people 2000 years before creating their bodies, and He created their sustenance 2000 years before creating their souls.
Allah (SWT) Has granted enough sustenance for all His creatures, but arrogant people snatch away the rights of others causing them to be hungry. Ameerul Mo'minnen Ali (AS) said: No one becomes hungry but only because someone else has taken away his right. ما جاع فقير إلا بما منع به غني
The real cause of the poverty is the injustice done by wrong doers on others.
You are responsible about your father. Do not miss this great chance to serve him in his old age and do not listen to any one including your wife if she wants to discourage you from serving your father.
Uttering falsehood is one of the major sins in Islam. Islam is a religion based on truth and therefore does not accept spreading of falsehood. In the Holy Qur'an believers are invited to "advice each other to the truth" (103:3). In the said case, it is incumbent for a Muslim to act upon the principle of enjoying good and forbidding evil. Its obligation is based upon the following conditions: the person enjoying good and forbidding evil should be able to distinguish between good and evil, there should be a possibility from stopping the transgressor, we should be sure about the persistence in sin by the person whose behavior should be rectified (meaning that he does not repeat the sin), and we should be safe from harms that may bring greater damages while performing this duty. If all these conditions are met, it will be necessary to enjoy good and forbid evil by advising the person uttering falsehood to rectify his behavior. If he does not listen to the advices it will be allowed to use stronger words, and if this does not work too, it will be permissible to take practical steps according to wisdom and circumstances. If all of these are not possible, and other people too are unable to fulfill this kifa'i responsibility, to reject falsehood within your heart will suffice insh'Allah.
With prayers for your success.