I pulled out the solenoid.  It has a little rubber cap on the top of the plunger.  I pulled the cap off the plunger and put the solenoid back in the carb.  The machine started right up and ran good for about 10 seconds.  Then it killed again and won't restart.  I put the rubber cap back on and tried again, but it won't restart.  

Did you  check the solenoid by watching the plunger, and powering it on and off?
 
It should retract and extend..
 
 
 
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I checked the solenoid with it pulled out so I could watch it.  Turned the key to all positions.  The solenoid did not move.  I would say it's dead. 

I checked on cost of parts.  Used solenoid is $10 on ebay.  $70 for new solenoid.  New carburetor is $125, including a solenoid.  Thinking back to the backfires that happened when the flywheels slipped - they were heavy backfires.  Not sure how the carb stands up to that.  I'm thinking I may try the used solenoid, if it works great.  If not, I'll get a new carb. 

Yes, it's dead...
 
I'd just cut that pindle off as short as possible so it does nothing.
It won't hurt anything and will never be an issue in the future. 
Your using the unit, just not the pin which is the issue...
 
As I stated earlier, if the engine is super hot AND your at WOT and just
immediately kill the ignition you may get a backfire.  Idle down for 10-15 seconds
and you shouldn't get anything..
 
The really bad backfire you had was because of the flywheel keyway being broke.
 
 
Definitly DO NOT put on an aftermarket carb for the solenoid.  Their a 5/50 crap shoot
and may very well cause other issues(that you don't currently have).
 
 Plz post back your ultimate fix..
 
 
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Hi Scott - I cut off the spindle and put the solenoid back in.  The engine runs now steady.  Except - it doesn't run real smooth.  Not sure how to describe it except saying it's a little bit rough sounding.  Not horrible though.  The bigger thing is the muffler gets SUPER red hot pretty quickly.  I shut it down and I don't feel like running it more than a minute since it gets so hot.  I'm glad the engine runs and stays running.  But, I'm concerned about the muffler.  Reading up about the muffler, seems maybe it could be running lean - which may relate back to the carburetor?  The other thing I read is maybe timing could be slightly off - but I'm counting on the 100 ft-lbs has the flywheel right where it needs to be.  I'm leaning towards doing more with the carburetor...

I agree, the SUPER red muffler isn't normal.
 
I would re-check valve clearances AND pull the spark plug and inspect it for it's color.
 
It should be burning tannish (do a search for pictures).  
 
Do you have the correct heat range spark plug installed (IE, not too hot)?
 
Have you been thru the carb looking for any crap / obstructions?
 
Do you have the NIKKI or Ruix carb?  NIKKI has the plastic internals...
 
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Do you have a fuel pump or is it gravity fed?
 
In any event, get a container and put the carb end of the fuel line into it.
Let gravity pull fuel or start the engine.  You should see a decent amount of fuel flow.
5-15 seconds is plenty of time to check for flow....
 
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