When asking Gareth about buying the shed there is a specific (in-game)system popup that asks you if you want to buy the shed. If you just give Gareth money he isn't likely to give you anything, unless you use that pop-up to buy the shed key. Though you can totally get the key from him without the system pop-up. Even get it for a good discount or for free if you know how or get lucky like I did a few times.
I believe you can't build structures out in the wild because the world is randomly generated every day. Presumably, anything you could build will be lost. You can only build a camp fire and place items like a tent if you're making a temporary camp out in the wilderness.