I let mine out all the time, actually I feel guilty if they don't get out on a regular basis. Sitting in a cage, no matter how pretty a cage, is still just sitting in a cage. One of their favorite hobbies is to sit on their perches in front of the TV. They love to watch movies, especially sci fi ones involving giant bugs going on a rampage and eating people.
I handle my bugs all the time, and have never noticed any negative side affects to my crew. Out of all my mantises, I've only had one mantis that seemed stressed by handling and that was my ghost Spook. I did try to respect his wishes and keep handling on as needed only basis since he was so timid.