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.

What is the ruling on tasting food while fasting?

Tasting food while fasting is disliked (makruh). However, if any part of the food reaches the body cavity (jauf), the fast becomes invalid.

Can someone break a make up fast (qada) after beginning it?

Does the clipping of nails nullify ablution?

No, it does not, and it is preferable to wash the hands afterwards.