Here, V must increase to keep P and n constant. n is the mole of the air inside the balloon, it cannot be changed as hot air doesn't leak out. With the constant n and larger V, its density is smaller, lighter.
I doubt about this statement. What you talk about here is the molar mass, it is rather different with density. If we consider the moist as a mixture of water(liquid) and dry air, apparently, the density of moist air will be larger.
I can also explain in your point, 1 mole of dry air occupies much larger volume than 1 mole of water. The differences in volume is much larger than the difference in mass when we consider the same quantity of dry air and water. Overall result is that density will be increased when water droplet is added.