For 1 & 2, you are correct. You want everything in the bath to be at the same temeperature, so that the temperature you record in the bath is the same temperature as your sample. Heating slowly and providing good circulation help the system reach thermal equilibrium.
For 3, if you don't dry your sample completely, you are introducing an impurity (water). How do impurities affect melting point?
For 4, you are correct. Air is a poor conductor of heat and can insulate parts of your sample from the surroundings, making your temperature readings inaccurate.