How should I be kind to my atheist father?

Allah, The Exalted, has decreed that we be kind to our 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.

What should the man, whose wife wasn`t on good terms with his mother, do?

Both the mother and the wife have rights on the man in question; therefore, each should be given her rights justly, and he should resort to wisdom to please both of them and make reconciliation.

Is everyone obligated to follow the fatwas of his country, and if we adopt the fatwas of scholars of other countries, is it considered a sin?

All perfect praise be to Allah, The Lord of The Wolrds and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.

It is permissible for the person who isn't specialized in Sharia sciences to apply the opinions of renowned scholars whom he trusts ,whether they be from his own country or not ,but if the scholars have different opnions regarding a certain issue/matter, then he must consult someone who is more knowledgeable than himself .It is preferable that you (the questioner) specify the case of your interest ,so that we could give you a more specific answer since some scholars deliver fatwas based on illogical/atypical opinions which should not be applied no matter what .And Allah The Exalted Knows Best.

Has Islam made it incumbent on us to give our children particular names?

No, it hasn`t, but it is desirable that we give them good names.