I'm a contractor in Maine and we certainly have a lot of experience with ice dams. An ice dam can be the quickest way to force water under the roof shingles even on a roof in good shape.
I have had mixed results with heat tapes. When they are used properly, they can be somewhat effective. Most folks expect to see a nice bare zig zag pattern along the edge of the roof with the ice gone around the tape. Not how it works!
If you carefully read all the directions on many of the competing brand heat tapes out there, you will see that they do not claim to remove ice dams. They are designed only to create a small open tunnel under the ice dam to allow water that is trapped above the dam to drain off the roof. This is great in theory but here is where the reality sets in. If the tape does not extend slightly beyond or below the drip edge of the roof, it won't work. If the top edge of the tape does not extend above the ice dam, they will not work. If the gutters fill with ice and bridge the ice to the roof, they won't work.
Many of the tapes instruct you to only operate them when outdoor temps are 30 degrees and above. I have found that they are ineffective in temps much below 20 degrees. They also caution not to let them run continuously.
Running them in the bottom of a gutter is almost a complete waste of time and money. Since they only are capable of melting a small area around the tape, a gutter full of ice will stay full of ice! If the downspout is blocked with ice, forget it!
What I have also found is that folks get a false sense of security when they have them installed and neglect to remove the snow from the bottom 3 to 6 feet of their roofs, figuring the tape is a substitute for roof shoveling. Obviously, when they are installed, shoveling or using a roof rake gets a little tricky, so as not to rip the tape right off the roof.
I guess you can tell I am not a fan of these heat tapes. One method I have found to be extremely effective with dealing with an ice damn are nylon stockings or pantyhose (cut into two sections) filled with salt, calcium chloride or potassium chloride placed over the dam. They last several days and melt slowly and make a nice wide valley in the ice. I have seen hundreds of gallons of water pour off from behind an ice dam using them. Chopping the ice dam can damage the shingles and can be dangerous, but the filled stockings are fairly quick and easy.
If you have plants below or are concerned about salt staining your roof or siding etc, I highly recommend Potassium Chloride which will not harm plants, grass etc. You can usually find potassium chloride at your local hardware store.
I can't take credit for discovering this method, but thankfully I learned it from a tip on This Old House many years ago. I usually make up dozens of them in advance, so I am ready for all the calls from folks who wait too long and call to see if I can come fix the water leaking through their ceilings!
From this picture it hardly looks like the stairs "cuts the room in half". In fact it looks like the stairs hug all the way to one side of the room.
I would strongly discourage you from proceeding with the idea that you propose. If the horizontal trap door were closed in the event of a fire it may very well be just that...a trap the prevents you from escaping in a timely manner, if at all.
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I know that in the USA sliding glass doors can leak if directly exposed to rain, which can be windblown.