These are nice and subtle questions.
Beginning with (3), there is a semantic difference between shop door and shop’s door. If I tell you I’ll wait by the shop door, then I generally mean at the front of the store (or maybe by the door for deliveries), but not, for instance, at a door that separates the shop from the living quarters. The same goes for shop window (usually, not just any window in the shop, but the display window at the front) and shop floor (usually, not the floor out the back where goods are stored, but the parts accessible to the public). By contrast, shop’s door/window/floor can refer to more widely. For instance, Today, I’m going to fix the shop’s floor might well refer to parts out back; and in a related vein, the shop’s side-window is fine, the shop side-window is odd.
Hence, in answer to (1), you use the possessive for a more generic meaning, and the compound to pick out some more idiomatic, conventionally salient door of the shop, etc.
With regard to (2), I’m not sure that ship crew is, in fact, unacceptable. If, in a hotel, we’re trying to stop sailors and pilots from fraternising, I might tell you: Put ship crews on the even floors and plane crews on the odd floors. If there’s a fight, I might ask: Was a ship crew that started it? When ship crew becomes odd is when we have a specific, known ship in mind—in the same way that it’s odd to say the dog owner, rather than the dog’s owner, when the dog in question is known and salient.