1992 kawasaki zx600. there is a little leak where the shifter is attached an sometimes it pops in to neutral while riding

The transmission is locked into park by a solenoid activated little pin that stops the shifter from moving. It is attached to the shifter area under the console. You say you replaced the brake light switch, but did you check the brake lights to see if they work?

the dash is notched were the neutral posisition is , this notch (shift lever needs moved out of notch) is there for two reasons , the one you need to be concerned about is the clutch disconnect , this makes it impossible to shift into any direction without clutch pedal down , sometimes this lever will become bent and not press far enough for release arm of shift lever to allow its movement

love these little tractors , they got all the features of a bigger machine but with the size ease of a compact machine , for a small farm or big home garden there's nothing this machine cant do with the right attachments on it I sure wish JD still made em !

If the bike just cuts out whilst riding and wont start straight away but will when it cools down its the pick up for sure.
You cant test it they only fail when hot

neutral light sensor on engine either corroded or needs replacing

Going into neutral has nothing to do with the oil. First, make sure the shifter lever isn't down too far. not allowing it to "return" after a shift.

Popping out of gear is caused by excess wear and/or damage to the transmission gears and/or shift forks. The gears engage each other using what are called "dogs". When the dogs become worn, they allow the gears to force each other apart - especially under accelleration - causing the transmission to pop out of gear. Repairing this mean a complete teardown of the bottom end of the engine will be needed.

The oil leak is a much easier fix; where the shift-shaft goes into the crankcase is a seal. Remove the retaining clip and carefully pry out the old seal and replace it with a new one.

Hey doesn't your Honda know that only Harley's mark their territory? Pull the cover and install a new gasket, no big deal.

is it diesel engine?if so chech ur heater plugs