If wallpaper was painted over you could tell pretty easily by pulling on some of the peeling paint and breaking the paint chips. Wear a respirator mask while doing this, however, as often times flaking paint is a potential indicator of lead based paint. If the chips contain paper, then you're right, it's wallpaper with paint over it. If it's just paint, then be more careful - get the chips tested for lead.
Given the wide spread flaking, it is likely that the wall simply wasn't properly primed. An improperly treated wall when painted over will eventually lose adhesion with the paint and it'll flake away like you're seeing. If they applied paint directly to wall paper w/o priming, I think the same is true.
That 2nd picture DOES remind me of wallpaper... I've scrubbed far too much backing off the walls and that looks similar.
The grey subsurface is, I think, a kind of stucco mix that was often used to even up walls where lathe and plaster was replaced with the older style 2x4 drywall panels. It's nasty, gritty, dusty, unpleasant stuff, tougher than joint compound/plaster to work with because of it's tendency to crack and break rather catastrophically. When I run into that stuff in my rentals my approach is, "IF I have to touch it at all, it's ALL coming down." Plus with wallpaper I swear gutting is easier than stripping.
Now around the vent pipe, that looks like moisture damage. The bubbling around the pipe suggest water leakage. Is that a "finished" ceiling - ie - thats the roof on the other side of that wall w/ the pipe? If so, make sure it's properly sealed and replace at least that area of ceiling.
Picture #5 seems to confirm this - someone touched it, and patched it badly.
Lightweight spackling compound is not suitable for exterior use. Even if painted, water will effect it greatly. In addition, it is prone to cracking as it is brittle and probably will not withstand opening and closing of the door (vibrations).
Wood filler or wood glue may work to fill the crack and would be better for exterior use. The glue is probably more flexible than the filler and may be better for filling the crack. I've used the wood filler before, but usually to fill nail holes or fill in small cracks. For a few bucks and some sanding, it may be worth a go round.
Another option would be to caulk the crack. Since caulk is flexible, it will resist cracking better than the wood filler. Both caulk and wood filler would be paintable. I don't think you can stain caulk, but I believe the door is painted on both sides.
The other option I mentioned in the comments is to replace the panel/door. The panel is most likely held in by surrounding trim, so if that can be removed, just the panel could be replaced.
Another option would be to get a very thin piece of veneer wood, cut it to the dimensions of the panel, and then glue it on top of the existing panel. Since you cannot see the crack from the outside, this could be done on the inside and then painted to match.
Best Answer
Scrap all the junk off the seam and wall, then sand it. Get some fiberglass mesh drywall tape and some joint compound and seam it. If you're not familiar with taping drywall, there are countless videos on the subject on the internet. Good luck.