I want to install a small propane range/stove in my cabin, but don't want the hassle & expense of a giant propane tank in the yard.
Propane appliances are not a problem, but the tanks are. Any propane cylinder or tank should be considered a potential hazard and not stored inside.
People often make exceptions for the small ~14 oz cylinders used for torches and portable stoves. Hardware stores will stock those small cylinders inside the store. However, tanks like the common 20 lb size used with gas BBQs aren't even allowed inside the store.
They're usually kept outside in steel or steel mesh cabinets that act as flame arrestors in addition to protecting the tanks. If you're talking about a 100 lb cylinder, that should definitely stay outside in a protected place away from anything flammable. Check out this advice on storage (tank position is also important).
You can pipe it into the residence, though, just as you would from a buried tank (the requirements are covered by code).
Is it safe and proper to run the appliance off of, say, a BBQ cylinder?...Are the pressures coming off a large, installed commercially-supplied propane tank the same as those from the regulator of a BBQ cylinder? In other words, will my appliance know any difference?
The gas, itself, determines the pressure. When the tank is closed, the pressure will be the same in any size tank. Some of the liquefied gas converts to gaseous form until the pressure is sufficient to keep the rest in liquid form. The size and shape of the tank might affect how quickly the tank can replace the gas being drawn off if you were feeding a monstrous requirement, but for typical residential use, appliances won't care what kind of tank is feeding it. You could use a 14 oz torch cylinder, but you would run out quickly.
I'm not familiar with the tank regulators and fittings that are required by code for different size tanks. Hopefully, someone else can address that aspect.