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 it permissible for a woman to sit with her brothers –in- law?

It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa(seclusion) with his sister-in-law.

Is it permissible for a woman to visit the doctor while in her `Iddah(waiting period)?

It is permissible for her to do that during day time, but it is impermissible for her to go out at night except for a sound reason.

 Reciting the Prophetic narration is permissible as many scholars used to, since in principle, the rulings of recitation have to be observed while speaking Arabic language.