Is it permissible for a person who has to make up for missed fasts to fast six days of Shawal(the month directly after Ramdhaan) before making up for the missed fasts ?
Yes, it is permissible for him/her to do so, but it is better that they make up for missed fasts first.
Is it permissible for the mother to give her children from the Zakah(obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
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.
Is it permissible to listen to chants praising the Prophet if accompanied by musical instruments?
If singing, or chanting in praise of the Prophet was accompanied by a forbidden musical instrument, then it is forbidden.