Yes, cold air is denser than warm air; one could also say it is heavier. We measure density as mass per volume (kg/m³ or lb_m/ft³) and specific weight (heaviness) as the weight per volume (N/m³ or lb_f/ft³). The difference is that the density of an object does not change if the gravity acting upon it changes, but the specific weight depends on gravity. For instance, if your mass is 60 kg, and your volume is about .006 m³, you have a density of 1000 kg/m³ (this is equal to the density of water, which is good since we can safely assume that humans are mostly water) and a specific weight of 9800 N/m³. The difference in these numbers is a factor of 9.8, which is the acceleration due to gravity on earth.

On the moon, you would still have a mass of 60 kg and a volume of 0.006 m³, but your moon specific weight would be about 1/6 of your Earth specific weight, because the gravitational acceleration on the moon is about 1/6 of that on Earth.

But returning to Earth and the question of the effect of temperature on air density, how can we explain your observation that colder air is denser? In my field, materials engineering, we study materials (e.g. metals, ceramics, semiconductors, organic materials like rubber and wood, and more) from the atomic scale (10^-10 m) all the way to the skyscraper scale (10² m), and we often find that phenomena on the atomic scale can have profound effects and explain the behavior of objects hundreds of meters long. For example, thermal expansion in a material occurs when the addition of thermal energy (rising temperature) causes atoms to vibrate with higher amplitude about their mean positions. This means that each atom takes up more space in all directions and is directly observable in the positive dimensional change (expansion) of most solids upon heating--or in contraction upon cooling. For example, the thermal expansion of metals used in satellites and spacecraft must be taken into account, because the temperature range (about 300 C) experienced by a satellite or spacecraft is large enough that it could cause a 100-meter long component to contract and expand over a range of 70 cm. I don't know about you, but I wouldn't want to find a 70-cm gap in my spacecraft! Here I have assumed the body of the spacecraft is made of aluminum, with a thermal expansion coefficient of 24 x 10^-6/deg C.

Thermal expansion is pretty important for metals, and the thermal expansion coefficients for gases are much bigger. For example, air at room temperature has a thermal expansion coefficient of 3.4 x 10^-3/deg C, meaning that raising the temperature of a volume of air 1 m on each side (1 m³) by 1 degree C, would expand that cube to a volume 1.0034 m on each side (1.01 m³). Remember, your volume is only .006 m³, so this is actually a big change! But remember, the mass of the air in that volume did not change, and therefore increasing the temperature of air means that the same mass of it takes up more space, and therefore it is less dense. Conversely, cooling 1 m³ of air air by 1 degree Celsius would reduce its volume to .9898 m³, meaning it would be more dense. You can see some data on the density of air with respect to temperature here.

Cold air is more dense than warm air. When the air cools, the molecules move closer together. Whether or not it is heavier depends on what you mean by “heavy.” Typically, cooling the air won’t change the composition, so it will be the same molecules with the same mass. However, it may appear heavier since the molecules are packed closer together- so a given volume (say 1 cubic foot) of cold air will weigh more than the same volume of hot air.

You are correct! Cold air is both denser and heavier, which results in lots of interesting and critical phenomena in the world. Because the air molecules in warmer air are moving much faster and colliding with other molecules much faster than cold air molecules, they naturally want to occupy a larger amount of space than cold air molecules for the same number of molecules. As a result, for the same volume of air, there will be fewer hot air molecules, making hot air less dense and cold air denser.

When you say cold air is heavier, it is important to qualify that the molecules do not change mass, but for the same amount of volume of hot and cold air (at the same pressure), that volume of cold air will contain more molecules and therefore, will be heavier.

For a more mathematical/equation-based understanding of what I am saying, you can look at the ideal gas law which relates the number of molecules and their temperature to their volume and pressure here.

As an addendum, the fact that cold air is denser results in wind across the world. As pressure differences between hot and cold air try to equilibrate, air molecules are forced to move around, creating wind. A larger pressure difference will cause more wind than a smaller pressure difference.

Yes. Hot molecules move around more and push each other apart. Read here.

All else being equal, yes, cold air takes up less volume, and is heavier than warm air. However, air that is under lower pressure will be less dense than air under higher pressure, and allowing air to expand cools the expanding air off, which means that air at higher altitude will be colder, despite being less dense, than air at lower altitude (this is why it snows in the mountains, but not on the coast).

Yes, if the pressure of the air remains unchanged, the volume of air does shrink with lowering temperature, hence the density of air is larger.