If you fill it up to the overflow, does it drain into the overflow or just fill that cavity?
After flushing the toilet run a little water in the sink. If that makes it go away, you have problem with the vent and you can put a cheater in.So it turns out the overflow is not hooked up. While the sink has never been filled (until now) to reach the overflow, obviously water can get back there in general. That said, there doesn't appear to be any smell coming from the overflow holes, rather only where the stopper in the drain is.