Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?

Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.

Is the one who doesn`t offer prostration of recitation during Quranic recital considered sinful?

Prostration of Quranic recital is a Sunnah due on whosoever recites a verse that contains a prostration. It is also a Sunnah for the reader and the listener. However, none of them is considered sinful for not offering it, but they deprive themselves from a great reward. It is obligatory to offer the prostration of Quranic recital in congregational prayer following the lead of the Imam.

What is permissible for the husband to do as far as having sex with his wife is concerned?

The couple are free to enjoy each other sexually provided that they don`t do it during menses. It is also forbidden for the husband to enjoy the wife from her anus.

Is it permissible to recite the Holy Qur'an or to play a recording at mourning gatherings? Or is it considered an innovation?