Buy a new door.
Unless there's something unusual about the door that's not evident from the description it literally isn't worth the time and effort to strip and refinish. Even if you price your time at $0.00 (which is fine, many do for home projects) there's a strong argument to be made that it's still not worth the costs involved, unless you ...
Stain remover worked fine for removing most of the stain and other upper lacquer/sealer layers.
A thin layer of veneer then was pressed on the door. All good without buying a new door.
Upload before/after photos, alongside details.
Update post after a year to report on the long-term effect.
Use stain remover, and chip a thin layer of wood, then apply a new stain. But I doubt that the pressed veneer is thick enough to allow for any thinning.
There's no fixing the existing veneer. Face veneers on plywood are very thin -- the internet says they're 1/30" on average, which means that half the time they're thinner than that. It might be a ...
Generally speaking, the exterior doors to your house are (or, at least, can be made) reasonably air tight. You'll want to use similar type seals for your door.
I'm thinking a product similar to this self adhesive gasket:
Image of M-D 20-ft x 1/2-in Black Door Seal Silicone Door Weatherstrip supplied by Lowes.com. No endorsement of product or vendor intended ...
In the end a normal concealed hinge from HAFELE did the trick. I guess a blum hinge would have obtained the same result. These kind of hinges have a decent range of adjustment so even if the layout specifies 9mm overlay, it can be easily adjusted to go from 8 to 10 and be extremely precise.
This is the one I went for:
Reading from the table on the right, ...