Jeannie mentions getting mail from Omar Kyahham but if she was imprisoned since 1st century BC she would not have known him because he was not born for another 1000 years in the 10th century AD from the time she was imprisoned.
The TV station is shown as KXIU-TV, which is a fictional station and was used on The Monkees (1965) and other productions, but a TV station in Florida would have call letters beginning with the letter "W," not "K." They should've used WXIU, which was used in Hazel (1961) and other productions.
At around 9 seconds into the opening scene, the camera switches to Jeannie's bottle where the smoke generating device becomes clearly visible, along with two attached wires, as the smoke representing Jeannie rises up.
The boom shadow is visible when Dr. Bellows and General Peterson are in the gym.