Warm air is more dense than cool air. Warm air is less dense than cool air Warm air and cool air have the same density

You are exactly right. The molecules in hot air are moving faster than the molecules in cold air. Because of this, the molecules in hot air tend to be further apart on average, giving hot air a lower density. That means, for the same volume of air, hot air has fewer molecules and so it weighs less

0.11 m/s and 0.28 m/s should be correct

it doesn't. it was imagined up in the middle ages.