Kilohertz is right on, sounds like you need new needle and seats. Viton tipped needles are worth their weight in gold!! 320 mains sound good. If you dont have a bearing in your power jet hose, put one in it or plug off the bowl and intake bell ports with a screw and some loctite. Make sure your main jet is at the end of the needle jet and not still screwed into the bowl.
For idle adjustment, you can use the drill bit trick, then fine tune each carb evenly to get to 1500 rpm. Then, loosen the slack adjusters on the top of the carbs. Make sure you have some slack, then slowly turn the adjuster out until the rpm's start to increase. mark the adjuster with a sharpie then back off one turn. Do the same to both carbs, then they are sync'd.
You can run without the air box for trouble shooting but I would avoid long wide open throttle runs.
As far as the over heating problems, you could have a bubble in the system. There is a bleeder under the seat you can crack open and get the air out. Run the sled at idle for a few minutes with the coolant cap off. this may bleed off some of the air. Also, make sure you have a snow flap on your sled. Without a snow flap, your not bringing enough snow into the tunnel to cool the heat exchangers and will over heat. On 87-90 Exciters, most aftermarket companies suggest to run without a thermostat to improve circulation. I run mine without as well. Make sure you warm your sled up thoroughly before you gas it hard or cold seizure could occur. I wouldnt recommend running without a thermostat if you are frequently riding in extreme sub zero temps though.