I don't like gore or violence in movies; I can tolerate a little, but my husband LOVES all the cop/detective shows. I can watch some (like NCIS - there was usually one death, then a lot of work around that; it didn't seem to bother me too much). Mostly not, though. My husband hates most of the stuff I like to watch - foreign films, romantic movices (both funny and drama), historical pieces. He also loves documentaries, which I normally don't like. We watch different stuff. Sometimes on the couch together with headphones or something. We have a few shows we both like - comedies, standup, and a few British shows - so we'll watch those if we want to watch something together. We don't have a TV, though, so it's pretty easy to watch different stuff on the laptops.

ETA: To your specific question, I don't have an anxiety disorder, though I do get some anxiety with my BP. The gore/violence makes me incredibly edgy in real time, and then the anxiety holds over in my day the next day(s). I watch sad movies/drama a fair bit. I'm incredibly sensitive to those, too. I prefer that to the anxiety, though; it's more comfortable for me.