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
Should nail polish be removed before making ablution?
Yes, it should be removed because it prevents water from reaching the nails.
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.
Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?
It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.