Without seeing your fireplace, I can make only a guess about the setup. If the fireplace is brick, it was probably originally a woodburning fireplace, and the gas unit was probably installed later. You should open the flue when you make a fire. Otherwise you will experience gas fumes coming into the house.

I am puzzled by your description of the gas logs as styrofoam. Are you sure about that? Gas logs are usually made of porcelain. Is it possible the styrofoam was put in place to protect the logs? They are delicate. If styrofoam is part of the protective packaging material, then obviously it should be removed.

If you close the glass doors and open the vents above and below the doors (here I'm really guessing) you will be creating a warm draft flow into the room. There may be a duct that starts at the vent below the gas logs, goes under the logs, then up behind the logs and then above the logs and ends at the vent above the logs. What this is supposed to do (I think) is warm the air that gets sucked in through the lower vent and then let it out through the upper vent. Many prefab systems are equipped this way and come with a blower to accelerate the spread of warm air into the room (although the blowers are often too noisy). If your system has no blower, it may still distribute warm air into the room through the process of conduction (because warm air rises), if it is designed properly.

In any event, you should probably have a certified chimney expert look at your system to make sure that any required maintenance is done. (I am not such a person and not promoting the business. I just think it's a good safety precaution.)

It sounds like you have vent free gas logs which are light weight and are designed to use with your damper closed. I bought mine from ventless-gas-logs.com. You should check the website to get an idea of what they look like. I would have described them that way because they do feel almost like styrofoam. I don't think you have vented logs as the logs would be very heavy in weight. In either case you will need to open the doors when you are using it and we often times open the damper a little to vent some of the heat especially if we are having a party and want to use them for atmosphere and not heat. We used to have vented gas logs but didn't get much heat from them so we switched to vent free gas logs. You should have the gas log set checked by a professional just to be sure they are vent free or look for metal tags which will give you a model number. If you have questions I would recommend calling this company as they have experienced people who know a lot about gas products.

Here is a link that might be useful: Ventless gas logs