In the manga and anime, Giovanni isn't Ash's father as far as we know. It's highly unlikely, though many fan support in this theory.
However, in "Pokemon Live!", the American produced musical, Giovanni is Ash's father. However, the musical is generally considered non-canon and so most people don't put too much stock in it.
Still disputed. In the manga he is not. Many believe that he is in the anime however. There is no sure way of knowing unless they say so on the show. I would like to believe that he is, however they did not recognize eachother in the second Mewtwo movie. But then again, movies are not true to the real storyline. So, again, there's no way to know.