does cold air rise at the same speed / rate as hot air ?

Answer 1

It is not the temperature of the air that matters but the density. There is warm air close to the ground and colder air above it. The warm air does not necessarily rise into the cold air nor does the cold air necessarily sink into the warm air. Why not? Because the warm air is denser than the cold air above it.

If you increase the temperature so that the air expands and a parcel of air becomes less dense than the air above it, that parcel will rise but it does so because of the density not the temperature.

Answer 2

No, cold air sinks because it is more dense.