It would help a lot if we knew what engine and carburetor you have. People usually assume it is a Chinese 2 stroke but we have no way of knowing that unless you tell us.
If we are talking Chinese engine, the "bleeder valve" you mention tells me you have the NT style carburetor. That isn't a bleeder but a primer of sorts. It simply pushes the float down allowing an over rich condition which sometimes helps on a cold start. If used, you just push t down and hold for a few seconds. It isn't a pump.
Does the engine shut down while riding or just when idling? There is a big difference. If the problem is at idle maybe all you need to do is adjust the idle speed screw in a turn or so. If the engine quits when you're riding you might have a weak ignition problem.
The CDI has been found to be a culprit when this situation shows up. They work fine until warm then start to fail. Rare but it has been reported several times.
You said you confirmed fuel flows from the tank. How steady is the stream when you check it? It must be a steady stream and not dripping or intermittent. Are you using an in-line fuel filter?
As far as the oil around the head, it could be the head bolts aren't tight enough or a bad head gasket. A leaking head gasket will often show up as oil and/or a ticking sound, especially when the engine is cold. If the studs are 8mm the nuts need to be torqued to 120 to 140 inch pounds. You might want to remove the cylinder head and inspect the head gasket. A leaking gasket is easy to see. Look for a dark area extending from the combustion chamber side to the outside diameter of the sealing surface. We also recommend that you replace the chrome acorn nuts with hex nuts. The acorns are notorious for bottoming out on the studs before applying force to the cylinder head.
Use the search feature and read about 'lapping cylinder head' for a better gasket seal.
Tom