In current English, sorrow is not used as a verb, though there are some older examples of this (e.g. from the Bible).
As for the difference between mourning and grieving, they are similar, but mourning primarily refers to the outward things that a person does when someone has died. So for instance, wearing black clothes or crying at the funeral.
Grieving is the emotion that a person feels when someone has died.
So normally the emotion of grief causes the action of mourning, but you can imagine a disconnection between the two. You could (hypocritically) mourn someone in public, but privately feel no grief at all.
It would also be perfectly normal to say, "Anna mourned her dad´s death." The preposition "for" is often omitted as Saratoga shows in the sentence, "You could (hypocritically) mourn someone in public, but privately feel no grief at all."