What is the ruling on seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is my husband entitled to take my salary?
Your salary is yours, and you may give some of it to him as a kind of free-will contribution.
Is it permissible for a woman to observe voluntary fast without her husband's permission?
A woman is not permitted to observe a voluntary fast (nafl) while her husband is present without his permission.