444 advice needed (sorry - long post)

Any air leak between the carb and the head will upset the fuel mixture and can cause a miss and hard starting. If the carb was a little loose when you were running the tractor that may have allowed the gasket to start to leak and cause the miss.

Since you know that you have gasket leaks that is where I would start the repairs. Fix what you know is bad first before doing anything else.

You can use jumper cables or another battery to get your timing light far enough forward to see the marks. If you are going to time it by ear you have to retard the timing some from where it runs best or you will be too far advanced and will risk scoring a piston.

The need for the spring on the needle and float is determined by the style of needle being used. If the needle has a Viton tip and a groove on the other end it is designed for the spring. The spring allows the float to help pull the needle open if it should stick in the seat. If there is no groove on the float end of the needle then there will be no spring. Even if the needle is designed for a spring it should still work but it may stick closed sometimes.

Unless it contains the main jet, the fuel cutoff solenoid only comes into play when stopping the engine. It is there to keep the engine from running on or dieseling after being shut off. Does your carb have an adjustment screw where the solenoid should be?

Does your 444 have an electric fuel pump? My 2444 does but it may be only on the industrial models. If it has a pump can you hear it run?

When you replaced the coil did you get the correct one? Even though you will not see a resistor on the coil, it is a 6 volt unit. The resistor is the wire from the keyswitch to the "I" terminal on the starter solenoid.