Leaning the bike to the side is the way to drain the rear oil. Then refill it with a pointed funnel carefully so as to not spill too much on the outside works and wheel. It doesnt take much.
Regarding your petcock, originally there was an intake tube assembly with a mesh filter on it. This enabled you to have a reserve and protected the carb against junk getting in it. You should also have a filter after the petcock and before the carb.
Those petcocks are generally available. Zippy in Ca has Vespa parts.