Now that the oil is gone and you are confident about the carburetor, let's start again.
1. Test the spark using the actual spark plug you intend to use to run it. We need to know that you are getting a steady stream of blue sparks across the spark plug, whenever you pull the starter.
2. Test the compression the Briggs way: disconnect the spark plug, tilt the mower so the oil runs toward the crankshaft, and rotate the blade plate backwards until you find compression. Then just flip the plate backwards, to see if it will bounce off compression and turn back a bit in the forward direction. If it will, it should have enough compression to run. Reconnect the plug lead.
3. Make the governor check I described in my previous post. This is important.
4. Remove the muffler, and remove the element from the air cleaner but put the lid-retaining screw back in.
5. Set the controls for start (speed control on maximum), verify that the throttle butterfly is wide open, prime the carburetor with the bulb as normal, and pull the starter a few times.
6. Report back.
Essentially, the "odd pop or two" indicates that it is trying to start. Its failure indicates that mixture, compression, or spark is quite a long way wrong. We need to find out which.
