As the others have said, your carb
is puking. Its caused by the heat of the engine, cooking the gas in the carb. When your riding, the air cools the motor, and takes the heat away. When you stop, there's no air circulation. In a stock configuration, there's no place for the expanding fuel to go, so it comes out of a carb vent. Some folks locate that particular vent, and route a hose off of it, to someplace below the bike. This helps part of the problem, but if somebody should happen to flip a lit cig under the bike at the wrong time!!!!!
Going pumpless, is something that people have had good success with regarding the puking problem. It's possible, that the fuel pump maintains enough pressure on the needle /seat valve in the bowl, to keep supplying fuel, as whats in the carb tries to get out. Without the pump, there is no extra fuel trying to get into the carb under pressure, to compound what would probably go unnoticed most of the time.
The brass colored elbow on the very bottom of the carb, ie the float bowl, is actually the carb drain. There shouldn't be anything leaking out of it. If this is the source, and it's not simply running down the outside of the carb from higher up(and dripping off of this nipple/elbow), then see if this screw can be snugged slightly(don't over tighten it. Snug is good!)
Another thought is what Davej above said, regarding the float level. If it is to high, there is extra fuel to expand, or it may be leaking out a bit at a time all the time, but I think you'd be smelling fuel all the time while ridding in this case.
Since it's a used bike, now is a good time to start getting acquainted with it. It's hard to tell who has done what in the past.
The scooter only holds a bit over 5 gallons in the tank, so if you get a total(including part of the reserve), of about 200 miles, without running out, you'll still be around 40 miles to the gallon at that rate.
Don't rely on the gauge completely. learn how much fuel is in the tank, by filling it to the same place all the time(bottom of the neck), and then seeing how much fuel it takes from say half a tank, or right when the reserve light blinks on. It'll give you a better understanding, of just what the gauge is really telling you(and how much fuel you really have, so you aren't sitting on the side of the road cussing the fuel gauge).
Welcome to RSC.
Enjoy your new bike, and you've come to the right place for RS