Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on the Friday bath (ghusl al-Jumu'ah)?
The Friday bath is a confirmed Sunnah, based on the Prophet's (peace be upon him) saying: "Whoever performs ablution on Friday, it is good for him, and whoever takes a bath, bathing is better." (Reported by Abu Dawud and al-Tirmidhi who said it is a hasan hadith). It is recommended for everyone who wants to attend the Friday prayer, even if they are not from its usual attendees, like a woman or a boy.
If a person is in a state of major impurity (junub) and intends by his bath both lifting the major impurity and the Friday bath, he achieves both. If he intends only one of them, he achieves only what he intended. And Allah the Almighty knows best.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
What is the ruling on one who slept while firmly seated on the ground, regarding nullification of ablution?
The default ruling regarding sleep is that it nullifies ablution. An exception to this is one who slept while firmly seated on the ground; his ablution is not nullified by sleep if he wakes up while still firmly seated on the ground. And Allah the Almighty knows best.