The little beasties sit at my door, waiting for me to open it....
Actually, they much more enjoy sitting on my patio furniture.
If they get into the house, they'll die somewhere and sort of dry out. Have found a deceased anole behind my desk and a live one in the wastebasket. If you want a screen for your sliding doors, call one of the companies that puts up bird cages around here, they can probably help or tell you who can. T&D, White's Aluminum come to mind for cage installers.
I do enjoy the little creatures, they are entertaining to watch, keep bugs on the lanai to a minimum and really seem to enjoy stretching out on my outdoor furniture. I don't mind, but they do leave a little "deposit" behind. I'm going to have to leave a scrub brush and cleaner for them to clean up after themselves.