Things I have replaced: points, condenser, fuel tank, fuel lines, fuel sediment bowl, wiring harness, main on/off switch, and the carburetor. 

Things I have tried: rebuilt the old carburetor, reset the timing, and used new fuel without the 10% ethanol.

The timing on this little engine is a bit confusing. Some say the number one cylinder is close to the radiator; others say it is close to the timing gears/crank pulley. It ran fine when I used the cylinder closest to the radiator for number one, but I am truly not sure which end number one is located. Is the number one cylinder located on the end with the fan belt or nearest the tractor seat?

Thanks for ANY and all help. 

I actually rebuilt the distributor a couple of days ago. (bearings, etc.)

This thing has been a nightmare for me. 

Any update? 

I see what appears to be new head bolts, but I don't see any notes of what else you may have done so far, so please consider that when reading what follows...

Flatheads have another peculiarity that'll cause this kind of thing...  carbon soot buildup on the valve stems... it cakes up, and as the engine warms, the stems bind in the guides, causing them to not close fast enough, not seal up tight... which causes manifold vacuum to drop, and makes the engine seem like it's running lean, starving for fuel.  Let it sit for a half hour, the stems cool, valves relax to the seats, and it'll start right up.

The giveaway?  When it stalls, turn it over by hand, and listen at the throat of the carb and exhaust.  As you come to TDC on every cylinder, listen for a hiss... that means you've got a valve leaking.  If it doesn't build much compression (or any) while you're hearing that, you've got sticky valve guides.

This isn't a 'difficult' fix, but it takes patience, at least, for the first engine you do.

Shut off fuel, Drain coolant, disconnect fuel and throttle/choke linkages, spray the head/manifold bolts with penetrant.  Take off the upper coolant hose, and... if you're NOT ABSOLUTELY CERTAIN there's no head valve under the thermostat, pull the upper neck and thermostat.

Unbolt the manifolds, swing the exhaust out of the way.

If there's a distributor coming out the top, you'll need to pop the cap off, loosen the retainer, and lift out the distributor... HOPEFULLY enough wire on the ignition that you don't have to disconnect it.

Unbolt the head... pay attention to bolt lengths, if they're not all identical, MARK the sizes for their respective holes, and WRITE IT DOWN!

Put on a mask and goggles, and scrub the cylinder head CLEAN.  Remember- this engine has been around a while, there's probably been some leaded gasoline through it, so the carbon soot you're breathing, will be an excellent way to get a little lead in you, so treat it with caution.  Put a good straightedge on the mating face of the head, see if it's flat and true.  If not, take it to a machine shop and have 'em touch it up.  Put a straightedge on the top of the block, and look for anything serious.  With the head flat, the block can be a little off and still seal up good with a factory copper gasket.

Remove the side covers (valves).

Take a sheet of aluminum foil, and wad up pieces to make plugs for the drain holes in the side covers.  This will keep STUFF from falling into the bottom of the oilpan...

Apply a spring compressor, and remove the valve spring keepers (a magnet and a pair of larger hemostatis is handy for this... if you drop one into the oilpan, you'll be angry that you didn't wad up some aluminum foil...

Now fold a piece of foil and place it as a cover over top of all the tappets... keep 'em crud-free for the next few steps.

Look closely at each keeper and spring.  If you see anything that is unlike the others, act on it accordingly.

Lift out one valve, and remove it's spring.  Place them in order as you do it.

Wire brush the valve stem, neck, and head....  I chuck 'em a cordless drill, and use a motor-mounted wire wheel to clean them up.  Once brushed clean, WRITE the cylinder and position on the head of the valve, then continue to next.

Once all done, place all springs on flat surface, look for oddities.  Short springs need replaced.  If you have a friend with a valve spring tester, have 'em all checked for tension.

Take scrapers, and a Dremel with wire wheel, and scrub the crud out of all the ports.

Use a valve stem 'cleaning reamer' (its a springy blade thing) and some penetrating oil to clean crud from the inside of each valve stem.  Check each guide with it's respective valve... it should fit well, and not be all sloppy, and not bind up.  If you have a bind, run the reamer a little more, testing frequently.  If you have big slop, pulling out a guide, dropping in a new one, and recutting the valve seats is a wise solution.  Seems serious, but it's not.

Get out your lapstick and lapping compound... put a little of the gritty paste on a valve's sealing surface, smack the lapstick to it, and spin it between your hands.  Feel the grit- it'll start out feeling roughish, and after a couple cycles, it'll start to smooth out.  Once it's feeling smooth, pull up the valve, wipe it, and the seat off, and look- you SHOULD see a nice shiny track where the valve polished itself into fitment.  Call it good and move to the next.

Once all done, loosen the tappet adjusters (two small wrenches), then install the valve springs with compressor and retainers.  Retainers won't WANT to just 'stay' where you put them while you fiddle, so put a dab of axle grease on 'em to help 'glue' them in place for the process.

Once all are in, you'll need to get feeler gauges and put a socket wrench on the crank to rotate... reset your valve clearances (should be in the engine's maintenance specs, usually 0.020-0.030" lash or so.  If you don't leave enough lash, the valves won't seat and seal... and when the engine is hot, the stems will expand enough such that the valves won't seat, right?

Don't forget to take out your aluminum foil before closing up the side covers.

IF you're going through this process, another thing that's good to do:

Pull out the head gasket.  If it's copper, it's PROBABLY re-usable, so try not to damage it... examine it closely for signs of leak-throughs.

Pull the block drain plugs, water pump, and any covers open to coolant.  Flush the block, and poke into all the water jackets with rods, brushes, porcupine quills, your wife's knitting needles, house wire, an old pair of crutches, a ski pole, Uncle Leo's fishing pole, chopsticks... whatever you can get.  Flush crud out 'till you can't get any more crud to come out.  Drain the radiator, flush it out... mebbie take it to a radiator shop and have 'em boil and pressure test it.  Consider rebuilding the water pump.  Look at the front crank seal, if it's leaky, pull that front sheave and get a new seal in it.  Sediment accumulates over time in these things, I have no idea why, but I've never found one that'd been closed up for 50yrs and didn't need a rinsing-out... sometimes, I had to chip chunks out... but ALL of them ran much better after I did.

Wire brush all the bolts clean, chase all the threads with a tap to clean them, Put it all back together.  On the head bolts that go through to water, I apply pipe paste sealant, as I've found that it solves LOTS of problems.  Torque the head using proper pattern!

Make sure you've got good seal on the exhaust and intake manifold faces, give the carbeurator a bath in the ultrasonic cleaner... (note that the fuel inlet fitting on many carbs has a little catch-screen IN it, and people don't know about it... so when it gets plugged, she'll starve.

This engine was used in several different applications. I believe the Cockshutt Pony used the same engine and I have seen industrial air compressors with similar engines.