motojoe

Take your thermostat out, it may not be working also bleed the system as you may have air cooling system.

Check your thermostat, make sure that it is not stuck, to check put it into hot water to see if it opens.

It appears that there is something wrong with either the carburetor or the fuel injection.  If it is carburetor engine then the needle and seat is not sealing or your fuel pump is staying on.  If it is fuel injected then your fuel pump is staying on when the engine is off.  Either way make sure that your fuel pressure is fixed before you run the new engine or it will fail again.

Look at the "Man Hole" threads and you will find a cut into the threads, line this up to the flywheel timing marks and that will be your TDC.

Just for fun if your compression is as low as you say check the valve adjustments a tight valve could cause the loss of compression and also cause it not to start.

Replace your stator and it should come with a pickup coil. that should fix your problem.

It appears that you are missing the gear sector (picture #2) on your new part.  For your new part you will need the "selector" that is made for your new part.  Why not clean up your old part?

Check out the "Run Low on Oil" sender; make sure that it is plugged in and works.  Sounds to me that something like the oil sender has a timer and is causing your machine to shut down.

Get the engine number off the engine block and Google it and it will tell you the manufacture of the engine.  Then look up the proper way to time the engine.

Your inter clutch (wet Clutch) needs to be replaced.

Check and make sure all your wires are connected correctly, one of the safety switch wires is no connected.

It appears that your internal "wet clutch" has gone out.

On the Flywheel make sure that you are using the correct timing mark and have it on the case timing mark then make sure that you have the camshaft aligned correctly.  Set your engine on top dead center and make sure the timing marks line up.                   

Does it have spark? If not then is most probably your stator.  If it has spark then it is either your fuel injector is clogged, clean it with carb cleaner until you have a good stream of fuel; take off your air filter and spray starting fluid into the intake manifold try stating it; if it starts then it either your fuel pump or your fuel line.  If it doesn't start then it is electrical (stator, ECU or a wire)

What is going on is your Stator's pick up coil is going bad.  When your machine is idling there is a lot of heat building up and your stator's pickup coil is loosing connection and the your are not getting spark to the engine.  Once the engine cools down the coil shrinks enough to make connection.  Long story short replace your stator.

If you got spark and fuel and assuming that the timing hasn't changed it appears to me that you have a exhaust valve too tight, adjust the valves.  If you have tight valves an engine has a hard time starting and yes when it is running you will get popping.  When you take of the valve cover you can check the timing as well but my guess is that your valves have tighten up.

I wouldn't worry too much about the valves or head damage as I have seen timing chains break and do no damage.  What you will have to do is check if the engine has compression and  if that is good (making sure that the valves are closed of course).  Then take off the valve cover (and maybe the head) along with the flywheel, stator etc. and replace the timing chain.

It sounds like your stator is going out, when the machine cools down it will run for a short time, and then shut off? When it shuts off check to see if you have spark, if not it is the stator or the ECU.

I think that you blew up the engine

It  just a crank case breather. You could do several things, put a small air filter on it (to keep oil from getting all over the place or plug it into your air filter box.

Bleed oil from the oil filter sometimes an air pocket or the filter does not have oil in it and will cause the light to come on.  To bleed the oil filter loosen the filter crank over the engine until oil comes out then re tight the oil filter, start engine.

You  probably have a stuck intake valve; adjust your valves.  Also your fuel pump is not working.

I would look at a Hisum or any other Chinese side by side that has the same engine size as yours their belt will probably fit, also look on e-bay as they sometimes give the belt sizes.  
As far as removing the belt just work it off by getting the belt off center and rotating the clutch.

Check the connection from the oil sending unit to see if it has a good connection and also check the ground.  If it happened just after an oil change make sure that you have the correct amount of oil in the crankcase, also sometimes the low oil sending unit has to have some time to refill after a oil change.

It is hard to determine from what you described but it could be that you haven't tighten the primary clutch down on the shaft and it is slipping on the shaft or the internal clutch is worn out.