That's good. It's not likely to be a mechanical problem making the chirp then. Sometimes they can make a noise "breathing." I have one that "whoops."
If you want you can check for spark now. It should be very easy to spin. I do think you have a spark since you say it backfired once.
That ridge at the top of the cylinder is normal.
Put the head back on. If you can, torque the head bolts to 190 inch pounds. Otherwise snug but don't lay on them.
It is very likely a carb problem. They're pretty common on a generator as they tend to sit with fuel in them then it gums up. That engine should have a float type carb. You'll see a float bowl under the carb.
How's the fuel? Is it clean? Disconnect the fuel line from the tank and make sure what comes out is clean and fresh. If it smells like bad (stale) gas, get rid of it.
When you're ready we can look at the carb.