Does touching one's wife invalidate ablution?
Touching the wife according to the Shafi'ee's doctrine invalidate husband's ablution. And Allah Knows Best.
Who has the right to child custody when final divorce takes place?
The wife has the right to child custody until her children reach the age of puberty. However, if she gets married, the right of custody becomes that of her mother`s.
Is it permissible for a wife to refuse to go to bed with her husband (for sexual intercourse)?
It isn`t permissible for her to do so unless for a sound reason.
What is the expiation for breaking fast due to being on a journey, or being sick, or being in a state of menstruation?
No expiation is due on the aforesaid categories, but they are obliged to make up for the missed fast. However, if any of them failed to do so while being able to, and the next Ramadhaan has come, then making up for those days is incumbent on him/her , and paying the ransom as well. And Allah Knows Best.