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.
Is it incumbent on a family to provide for its old handicapped son and take care of him?
The family and relatives - e.g. brothers - of such a person are obliged to spend on him and take care of him if he didn`t have money of his own.
Is it permissible to openly announce the engagement of a woman who is in her waiting period (iddah)?
It is not permissible to openly announce the engagement of a woman in her waiting period (Iddah), but it is permissible to hint (imply) at it if she is in her waiting period after the death of her husband. And Allah Knows Best.
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.