Is cold air heavier than warm air?

Air has mass and density. Yes, cold air is heavier than warm air. Cold air sinks because it is denser compared to warm air.  On the other hand, warm air rises because it is lighter.   When the air is hot or warm, its molecules are much further when it’s cold, its molecules are slightly packed closer which is why it moves slower.
The best association to this is the hot air balloon.  Hot air balloons fly because they are filled with hot air. Hot air is lighter than the cold air outside because when it is heated up, it expands making it lighter.  

If you have equal volumes of cold air and warm air ... two quart jars,
two gallon jugs, or two 1-cubic-foot balloons full of each ... then the
sample of cold air is more dense and therefore heavier than the
equal-size sample of warm air.