>>>It will generally start 3 or 4 times but never run for more than 30 seconds. After that, it will not start so I assume that eventually I would flood it by hitting the prime too many times.
If it starts then runs it’s not flooded from 3-4 primes. Some primers dump in a lot on one push and others only a small amount. You can check for flooding by pulling the plug.
>> > After it runs for 15 seconds and then stalls, it will not start up again unless I prime for 3-5 times.
This could be just a dirty carb.
>>> Once it is running, I have tried to spray starter fluid into the throat but that does not help and seems to have the same affect as giving it some throttle. It seems to stall the engine out. I have not tried to hit the prime when it is running.
You have to work the primer just right with slow steady primes to get the amount right. Keeping it running with a spray can is a bit harder.
>>>I was concerned that when I cover the throat with my hand, there does not appear to be much suction from the throat (i.e. air flow from air filter removed). If there is not a good air flow, would that explain why the engine is not running?
Yes, could be.
>>>How do I check the valve lash?
Take off the head cover and check with a feeler gauge. There are videos on utube.
>>> Also, how do I perform the leakdown test?
A dual pressure gage set reads air in on one and pressure in the engine on the other. A good engine will read 95% on the engine side compared to the air fed in. The %5 leak is expected. The more the leak % the more questionable the engine. Yes, you listen around for where the loss is.
>>>It sounds like I'd need to know when the piston is in the compression stroke and then forcing air into the spark plug hole. Would that have to be pressurized air to hear for a leak threw exhaust or carb?
Yes, also head and oil fill.
You description is a bit different than at first. You could have just a line block, needle seat block, needle seat problem or other area dirt. Check your carb linkage that the butterfly is opening fully when starting.