I would not use the live rock in your DT. you can probably clean the airstone good enough, but with how cheap they are, why bother?
I use cheap throw-away airstones for use in my QT tanks. I never use any live rock, sometimes I will use a bit of matrix in a bag in a HOB in the QT. I don't reuse that either, which is why I only use a little bit for my QT tank and then toss when needed.
PVC and sponge filters are ideal for QT use. quick and easy to clean and reuse. Keep the live rock in your DT, its too expensive to waste via using in QT.