All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
With regard to the first question, we say that the cousin should be advised in a wise and descent manner. The importance of prayer and the seriousness of being negligent with it should be emphasized for him, as well as the seriousness of violating the religion by listening to music, mixing with women in a forbidden manner and so on. If he repents and does what is right, then you can give him your sister in marriage, otherwise no. For more benefit, please refer to Fatwa 36805.
With regard to your second question, we appreciate your commitment to obey Allaah and avoid His anger. May Allaah reward you and make you firm on His Straight Path until the day you meet Him. Speaking to non-Mahram women is not permissible without necessity and provided one is safe from temptation as we have clarified in Fatwa 10463.
You must not obey your mother in her request that you speak to her female friends unless she will be caused harm if you disobey her concerning this and unless you are safe from temptation. For benefit about the limits of obedience to parents, please refer to Fatwa 84942.
No one is free from temptation, even if a woman is advanced in age and is like a mother. An-Nawawi said in his commentary to Muslim: “A woman is a likely object of desire and lust even if she is older; and they used to say, One man's trash is another man's treasure.” [End of quote]
If your mother's job involves matters that violate the religion, such as seclusion with non-Mahram men and buying things forbidden for them, then it is not permissible for her to continue with it; she has to leave it and look for another job. If she leaves it for Allaah's sake, Allaah will probably provide her with work which is better for her in return. Allaah says (what means): {And whoever fears Allaah – He will make for him a way out. And will provide for him from where he does not expect.} [Quran 65:2-3]
Also, the Prophet, sallallaahu ‘alayhi wa sallam, said: “Indeed, you leave nothing for fear of Allaah without Allaah giving you something better than it.” [Ahmad]
Allaah Knows best.