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.
When is a child instructed to fast?
A child is instructed to fast at the age of seven if they are capable of fasting and have reached the age of discernment, by analogy to prayer. They should be encouraged but not forced, so they can become accustomed to it. It is obligatory for their guardian, whether a father or another, to instruct them.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
Which is better for a traveler: fasting or breaking the fast?
Fasting is better, unless it causes hardship—in that case, breaking the fast is better.