If you are Dq'd you are unable to drive. I have seen where they told a guy that he was not making it as a driver, but if he wanted to he could do air driving. Pretty much said, we will DQ you or you could get more experience air driving and come out again next peak/summer.
Yes he can still drive on Saturdays. He would go back to what he was doing before he went FT driving. We had several drivers go full-time, and then they were sent back. They all went back to their PT positions, air driving or in the building etc. Unless he had an serious accident, I see no reason why they would stop him from doing Saturdays.
Yes, There is an Local sort guy / Sat Air driver that didn't make his 30 day and he still drivers Sat air in my area. However that probably varies by area and the people that are in charge.
Most people get DQ'd because they can't even get close to running scratch. Deliver 150 stops and 300 pieces is a lot different than deliver Sat Air...