How much will a damper clamp increase my energy costs?
Simply, you are not authorized to use the fireplace if you are ever going to forget to open the flue. If you can start the fire with the click of a remote I'd highly suggest leaving it open if you think you might forget. Perhaps a motorized flue control is in order (ideally one with limit switches that deny gas flow until satisfied-open). A CO detector goes without saying I'd hope.

Offhand, I can quantify for you that almost every fireplace installed in a house, has a net loss for heat. Including some even while they're in use. Keep the flue and doors closed when it's not.

houselogic.com will answer your bonus question by making sure you have a chimney cap present which keeps animals, rain, and snow out of the chimney.