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

Is it permissible to give the expiation due on the vow of anger to one person?

The expiation for the vow of anger-and that of the oath-is to be given to (10) needy persons, or a needy family of ten individuals, but giving it to one person is impermissible.

What should a sick person, who is incapable of fasting, and who didn`t make up for missed fasting days, do?

One who has missed fasting days is obliged to make up for them, but if he/she wasn`t able to because of an incurable disease , or old age, then he/she has to pay a ransom which is feeding a needy person for each of the missed fasting days. And Allah Knows Best.

How should I be kind to my atheist father?

Allah, The Exalted, has decreed and ordained his slaves to be kind with parents, even if they are polytheists, by honoring and obeying them in all that isn`t sinful, so that they might love Islam and embrace it as a result.