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.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
Is Iddah (waiting period) incumbent on an old woman whose husband had died?
Upon the death of the husband, Iddah is incumbent on the wife regardless of their age. And Allah Knows Best.