Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
Is it permissible for the man to go out wearing shorts?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is permissible so long as his Awrah (private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed. And Allah Knows Best
What is the Islamic ruling on parents asking son to divorce his wife?
Praise be to Allah the Lord of the Worlds.
If their asking him to do so is justified in Sharia, then there is no blame on him to obey them. And Allah The Almighty Knows Best.