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.
I think you missed a really important step in the process. Anytime I have to remove paper from walls that were not treated with a release sizing before the paper was applied, the drywall paper is usually damaged or lots of glue residue is left behind. I have found that a skim coat of drywall compound needs be be applied, sanded and sealed with PVA primer before painting can be done.
In most cases, a self priming paint is only effective on previously painted walls, but never on new or bare sheetrock. The primer in those paints is designed for hiding old color, not sealing new sheetrock. New rock, skim coated rock, etc., need to be sealed with a quick drying sealer like PVA primer. This type of primer seals the porous surfaces and prevents the moisture in regular paints and primers to soak into the drywall.
The skim coat is actually the key step you missed. do not use untreated regular drywall compound for this job. Modify the mud with a bit of water and some Ivory dish liquid. A gallon of mud with an ounce or two of water and a couple of tablespoons of Ivory is a good formula. Mix well. It will be loose, almost runny. You will find it flows on very smooth without a lot of bubbles with a 8 to 12 knife or trowel. After it dries (8 to 12 hours), sand lightly with a 220 grit sanding sponge. Touch up any areas that still need attention, dry and sand. Clean off all dust with a slightly damp sponge, let dry then seal with PVA primer. You can usually paint in about 2 hours after PVA.
As for the caulking you mentioned, remove as much as you can with a stiff one inch putty knife. Mineral spirits will usually remove remaining residue from the wood cabinets. Remove all of the caulking from the drywall with your knife, even if it takes some of the drywall paper with it. You can repair the divots in the rock with the skim coat as described above.
Best Answer
A good sealing primer is all that's really needed unless you have checking. In that case, a good wood filler will do. An oil-based primer might provide a more durable base and fill any fine cracks better.