Right now I have in my den a 110" Da-Lite HP screen on the north wall. Next to it I have a 42" VIZIO flat screen. I usually play them at different times - the projector at night and the flat screen in the day light - but I have had them both on at once. They don't interfere with one another in any way that I can see.
If you want to watch a different TV show from that of your wife there are easier ways to do it. You could simply record one and watch it later or you could put a flat screen in another room. Two people watching two different shows in the same room sounds loony.