What are the proofs that grant the father the right to name children, and what about Mary's mother naming her daughter "Mary"? Was Mary's father alive when her mother named her?

The person with the most right to name a newborn is its father, because the child is attributed to him and called by his name, as indicated by verses [of the Quran] and [Prophetic] hadiths. This is a right for the father, even if he is not obliged to spend on the child due to his poverty, then [the right passes] to the grandfather.

As for Mary's mother naming her daughter, it is probable that her father had passed away before her birth, or that she named her with his consent.

Spouses are advised to show mercy and understanding towards each other when naming children, in order to avoid dispute and preserve affection.