Oil coming from behind carb:
If the mower sat a long time without use,the rings may have seized to the grooves in the cylinder,thus not making a good seal between the cylinder wal and piston.
If this is the case,some of the combustion pressure seeps by and creaytes a positive pressure in the crankcase.When this happens the crankcase becomes a pressurized oil reservoir that oil will make its way out of to any opening with a lower pressure on the other side.(As in the crankcase breather tube in the air cleaner box). In laymans terms it is called blowby which may ultimately require a new set of rings on the piston as long as the cylinder bore is in a good condition to accept new rings.
As far as running a bit and shutting off,the float needle may be varnished from old gasoline and not allowing a sufficient flow to the carb to maintain engine operation.