Matteas,
If, when you turn the key off, the engine shuts down, the key is wired up so that it routes the output of the charging coil in to the shutdown selenoid, and connects the ground directly, instead of through the low oil switch first. So if you turn the key to off, with the engine turning, it trips the shutdown selenoid. If you stall the engine first, then turn the key off, it wont trip the selenoid.
I'm really surprised about the starter. I've never seen these engines act that way, unless the battery was really low. Lead acid batteries supplied by the Chinese companies have a bad history, I ususally use a garden tractor battery from Walmart. More capacity, and much better shelf life.
Steve