I can answer your side question: The Saudi Arabian flag was created after the Saudis took over Riyadh in 1902, when a sword was added to an older Wahhabi design of a green field with the shahadah on it. The flag was only used in Nejd at first but became the flag of Saudi Arabia when Hejaz and Nejd were fully unified in 1932. However, this design wasn't quite standardized, so the elements were drawn differently on different flags. The sword was usually curved instead of straight, there were sometimes two crossed swords instead of one, and before World War II flags were often reported with a white stripe down the side. The standard design with a single straight white sword and a relatively small inscription was established in 1973, and the most recent official drawing was done in 1984. As far as I know there was no consistent calligraphy before then.

As for your main question, we don't know who did the current calligraphy. That kind of information is usually only preserved in exceptional circumstances (i.e. Saddam Hussein's handwriting on the old Iraqi flag). I believe Ibn Saud himself is thought to have added the sword to the flag in 1902, but there's no reason to think he ever personally drew anything.