My stbd engine squeels at idle a lto after running a long time, it comes and goes, just needs to be lubricated, I forget the exact location, my dealer did it when I wa sin for service, mad eit better but it came back over time. I've been told it is normal and not going to harm anything and just to leave it alone until it breaks...mines been doing it for 3 seasons now...about 400 hours so I am not overly worried, just more annoying then anything else. Luckily I have the well dam and that blocks out 90% of it, only hear it if I'm in the cockpit rigging as someone else drives out. As far as it being tied into the belt, it should not be.
Overheating at idle means they might have installed the pump impeller upside down, it will grind a little and not turn well and will not circulate enough water, when the thermostat opens when in gear and a few rpm's higher, more water passes through the engine cooling it down which might be how the alarm goes off. You have not stated how the water feels...do you feel a difference in water temp, is it notiable? It could be something else if the water is still cold but the alarm triggers. Are your intakes in the lower unit free of growth? at higher speeds you could be forcing in water which allows it to cool, but the grwth could be blocking enough water from passing through at idle when sitting.