Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
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 shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
I missed fasting 30 days of the obligatory fast within the last 6 years and 20 days. How do I fast? And what should I do?
Missed fasting days must be made up, and it is permissible to make them up on non-consecutive days, as continuity is not required for making up missed fasts. Additionally, expiation (Fidya) must be given: feeding one needy person for each day if the makeup is delayed for one year. If the delay extends to two years, Fidya must be given to two needy people per day, and so on. And Allah Knows Best.