HVAC questions

Furnace in basment, AC unit outside. All up stairs rooms have intake and outlet vents. Basement has only outlet vents and are closed.
1. When AC is running and upstairs doors are closed, there is draft coming from each room under the door. When basement door is closed it has a huge draft from under the door leading downstairs. Is this normal? I would think that this is sucking my cool AC air and make the AC, running longer. I put a draft sock at the base of the basement door to help. But my cats need to get down there to use the litter box and it is a pain to keep basement door closed.
2. I thought about cutting the exposed intake duct in the basment and putting 2 6"x12" closable vents to suck cool air from the basment so I can cool the house without using AC all the time. Will this be a good idea?

Thanks, Nathan