Is it disliked to clip one`s nails at night, and to burn them afterwards?

It is recommended to bury hair and nails that have been removed and to conceal them in the ground. As for burning nails, we have not found any specific prohibition against it, but adhering to what is supported by evidence—burying the nails—is preferable. And Allah Knows Best.

Is it permissible for a person to give the Zakah (obligatory charity) to his daughter-in-law?

Yes, it is permissible for a person to give the Zakah to his daughter-in-law if she was poor, and none provided for particularly by whom are obliged to provide for her. And Allah Knows Best.

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 for the woman to use her mobile phone while on the street?

All Perfect Praise is due to Allah, The Lord of The Worlds 

Modesty should be observed by the woman at all times, so she is free to use her phone while on the street as long as it is done within the limits of Islamic manners. And Allah The Almigty Knows Best.