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 not fulfilling a vow?
A vow is a commitment towards Allah, The Almighty, and so, it should be observed as much as possible, and it is impermissible to take it lightly. Allah, The Exalted, Says –in commending the believers (What means): "(Because) they perform the vow and fear a day whereof the evil is wide-spreading." [Al-Insan/7].
Should vowed fast be offered in consecutive days, or not?
Vowed fast should be offered in line with what was originally intended by the vow-maker, either consecutively, or not.
Is it permissible to fast on behalf of the dead who died owing missed fast to make up?
His relative should fast on his behalf, and it is permissible for the guardians to give permission to non-relatives of the dead to fast on his behalf as well. And Allah Knows Best.