No, you shouldn't always use the plural.
This is easier to explain after answering your second question:
Yes, the singular 'store' can mean either a stock of materials or a shop.
You should use the plural form when you're referring to something that is plural, whether it is multiple shops or multiple stocks, and the singular when referring to a single thing, whether it is a single shop or a single stock.
Addressing your examples
cupboards for medical stores
A ship would likely have stores of multiple different types of medical materials: at least one store of bandages, at least one store of painkillers, at least one store of antibiotics, and so on. Together, these are the "medical stores". So you use the plural because you are referring to more than one store.
he keeps the ship’s stores
This is like the last one, but all kinds of different supplies, not just medical ones.
crates started arriving at the quartermaster’s stores
This could be ambiguous if you don't know what a quartermaster is.
If it said "the tailor's stores" you would probably assume it meant that crates were arriving at the shops owned by the tailor, as tailors tend to sell things.
Quartermasters are in charge of keeping track of supplies. If you're in the military and you need some shovels to dig a ditch, you might go ask the quartermaster for some. So, the quartermaster's stores are just where the quartermaster stores all the different supplies. The spot where each type of supply is stored is a store of that supply, so we have multiple stores.