Can I play a DVD and use Airplay mirroring to watch it on my TV ?

Welcome to the Apple Community.

Yes, but not with Apple's built in DVD player, you will need to use 3rd party DVD software.

Are you completley sure about this. AirPlay blocks copyrighted video content. So things like HBO2Go and NBA content can not be played over it. I am betting they block DVD conent as well.

Yes.