Sounds like the ignition isn't working sometimes. I would check out any switch on it that controls the ignition. Look for loose, corroded or frayed wires at the switch. If you find a loose wire, fix it and it still does the same thing, it might be in the switch. I might for test purposes connect the wires around the switch and see how that works. Of course, you'll have to cut it off the old-fashioned way while that switch is out of the loop - wear heavy gloves and pull the spark plug wire. :) Maybe you can get a new switch or adapt another one to work. You're lucky that old rascal still runs at all. Most likely, repeated starting attempts flood it, which would account for the rough running when it does make contact.

It's also possible that your plunger is worn . That mower uses a pump type plunger that has an o ring at the end to provide a seal. Pumping ten times is a lot. It's also possible that you are flooding it. Try using fewer plunges and see what happens.