On the cooling system side of your question, check these places first, there is a fresh water flushing device on the motor it is normaly on the port side towards the front under the lower cowling, a gray hose with a fitting on the end for attaching a garden hose, it has a threaded plastic bracket that you thread in onto when not in use, be sure the washer is in place inside the fitting and that the fitting is tight and there are no cracks in anything.
Check the small hose from the engine to the outlet hole (tell tale, or water pressue indicator hole) in the bottom cowling for blockage. If the motor is not overheating it could just be clogged up.
On the back of the motor, you will see a large black hose running from the exhaust plate between the heads, to the powerhead adaptor plate under the powerhead, the hose leads to the pressure relief valve in the exhaust plate, remove the cover and inspect the valve, seat and spring.
As far as the gearcase jumping out of gear, check your cable adjustment at the point where the cable end attaches to the motor (the shift linkage at the "slider) you should see a mark on the slider and the bracket (a line on both parts) they should line up in neutral. If they do not line up with shift handle in neutral then adjust the cable by moving the threaded cable end in or out to line the hole up with the pin. (YOU MUST HAVE AT LEAST 8mm OF THREAD INGAGEMENT OF CABLE END) also tighten the lock nut.
If they do line up (Shift handle in neutral and shift slide in neutral) and the prop does not spin freely in both directions then the shift shaft was not lined up properly when the gearcase was reinstalled. maybe one spline off.Jumping out of gear under load is a bad thing, can damage the gearcase, you need to drain the lube and look at the fill plug magnet for metal Fuzz is o.k. that indicates normal wear, but chunks or flakes means damage. If the gearlube has sparkles in it , or looks gray that is a sign of metal being shed from gears or clutch.
Marine Mechanic
Mercury/ Mercruiser Certified/Verado/DTS Yamaha Two and Fourstroke Certified 26 years experience