Yes, from the kinetic theory of matter and especially the kinetic theory of gasses, the higher the temperature (T), the higher the speed. The relationship scales approximately as 1/mv² ~ 3kT … so the temperature is proportional to the square of the speed.

As far as hot vs cold water, water freezes at 273 K or 0 centigrade at 1 bar pressure. Then, it is POSSIBLE to start with cold water say 10 C and hot water say 80 C and IF T YOU CAN ARRANGE TO TRANSFER HEAT C VERY FAST FROM THE G HOT WATER and sufficiently slow from the cold water, that the hot water can be brought to 0 C in less time than the cold water. It is all a matter of heat transfer rates.

I you hold the heat transfer rate the same, then the colder water will get to 0 C sooner than the hot water; it is all a matter of the rate at which heat is extracted from the water, it is a transport problem. The freezing point of 0 C at 1 bar pressure is however a thermodynamic property of the system and has NOTHING to do with rates.

I cannot think of a reason why hot water would freeze faster than cold water, if the two systems are positioned in identical conditions. About your second question on why things expand when heated, your second statement is correct. In fact, temperature is not even so well defined for a single molecule, it is defined for a macroscopic number of molecules, and it describes the average motion of these molecules. So when the temperature is higher, molecules move faster, and on average each of them takes up more space.

Regarding your first question: it is possible, but the conditions under which it happens are not well understood. I had actually not heard of this before, but apparently this phenomenon is called the Mpemba effect, and it has been a curiosity for a few decades now. There is a great link here about the many experiments that have been done to try to figure out what is causing this effect: Mpemba effect

Regarding your second question: It is generally true that molecular motion speeds up at higher temperatures. This in turn increases the average distance traveled by that molecule before it bangs into another molecule and changes direction (a quantity called the mean free path). This means the molecules tend to travel further and as a result there is more space in between molecules, so the same number of molecules will spread out more. But the molecules themselves don’t change size as far as I know.

Hot water should not freeze faster than cold water, given that all the other conditions are the same, i.e. the amounts of the hot and cold water are the same, the containers are similarly shaped, and they’re allowed to freeze in the same freezer with constant temperature, and so on.

It would be easy to test whether hot water freezes faster by putting the same volume of hot and cold water in two separate but similar containers, placing both in the freezer, and checking them periodically to observe the progression of freezing, but please leave some room for the water to expand so that the containers don't burst.

As for volumes - the volume of the molecule itself will not become substantially larger with temperature increases, but as you said, because the movement speed of the molecules increases, each molecule will need more room in which to move, and the overall volume of the liquid will therefore increase as long as the container is not rigid and/or the liquid does not take up the entire container. The overall volume of the liquid increases because faster molecules push away from one another than slower molecules, not because the real volume of each molecule increases substantially.

In answer to your first question, yes. Hot water has been observed to freeze faster than cold water, an effect noted by such famous scientists as Aristotle, Francis Bacon, and Rene Descartes. A few tough-to-prove explanations have been proposed for this, for example, since warm water evaporates faster than cold water, and evaporation is endothermic, warm water loses energy faster and cools more quickly than cold water. Another explanation is that water's hydrogen bonds--intermolecular bonds which are weaker than the covalent bonds between the 2 hydrogen atoms and 1 oxygen atom that make up a single water molecule, but stronger than Van der Waals bonds which rely on the continuous fluctuation of electric dipoles in molecules that are very close together --are shorter in cold water, meaning they hold water molecules with their covalent O-H bonds closer together than in warm water. In warmer water, both the hydrogen bonds and the covalent O-H bonds stretch out and relax, meaning they hold less energy, which is equivalent to cooling down. In this view, cold water has more energy to lose in every molecule, so naturally, it will cool more slowly than warm water.

As for your second question, you allude to the fact that an object's temperature is a measure of the average kinetic energy of molecules inside it. Kinetic energy is the energy of motion, given by 1/2 mv², where m is mass and v is velocity. So the molecules in a warm object are, on average, definitely moving faster than those in a cold object --we've just said, they have a higher velocity. However, this does not always directly translate to taking up more physical space. We address this in terms of a coefficient of thermal expansion, which is a material property, meaning it has a specific value for specific materials --like copper, glass, or silicon. In some cases, you're absolutely right, warm objects get slightly larger. Such materials have a positive coefficient of thermal expansion (think of running warm water over the metal lid of a glass jar in order to open it--you are taking advantage of the fact that the metal has a larger coefficient of thermal expansion than the glass!).

I think you know where I'm going with this: some materials actually shrink as they heat up, which brings me back to our good old pal, water. Solid ice floats atop liquid water. If water near 0 degrees Celsius had a positive coefficient of thermal expansion, liquid water would be less dense than solid ice (due to expansion), and the ice wouldn't float. We know that ice floats, and water therefore expands when it freezes. This can only mean that (for this temperature range), it has a negative coefficient of thermal expansion. As it turns out, water reaches a peak density around 4 degrees Celsius, and beyond that it becomes less dense (expands) as it is heated. Water is pretty mind-blowing, right? Thanks!

Hot water will never freeze, period, because the kinetic energy of the water molecules will overcome the electric dipole interactions that the water molecules have with each-other that allows them to form crystals.

Yes, hot materials expand (usually) because the molecules are bouncing around more vigorously and so take up more space. There are exceptions. Water and ice are in fact one of them: the lowest-energy distance between water molecules is actually larger than the closest space that you can pack them when they are liquid, which is why ice is actually less dense than liquid water.

Your intuition about hot water freezing faster than cold is a good one, however we need to carefully define what we mean by "freezing faster". It is true that hot water cools at a faster rate than cold water, but it is certainly not true that hot water freezes faster than cold. To see why this is true, we note that the cooling rate of any liquid is a function of temperature.

In general, the farther away in temperature the liquid is from the average temperature of the system, the quicker it will move toward that average temperature. In the case of freezing water, the system might be an ice tray in contact with the cold air of the freezer. The freezer air is at some average temperature, and the water will cool toward that temperature. The warmer the water, the more it wants to cool down. This is also true for the opposite process (i.e. boiling water). By the same logic as above, colder water will heat up faster than hotter water will.

Now that we've established that a liquid will move toward the average system temperature (and will do so more quickly the farther it is away from this average), we can think about why this still does not mean that hot water freezes faster (or conversely, that cold water boils faster). Say we have two baths of water, Bath 1 at 10°C and Bath 2 at 20°C. To freeze these baths, we must bring their temperatures down to 0°C. Bath 1, starting at 10°C, will cool at some given rate and freeze in a corresponding amount of time. Bath 2, starting at 20°C, will cool at a faster rate to begin with, since it is hotter and therefore farther away from the freezer temperature. However, in the process of cooling to 0°C, we note that Bath 2 must at some point reach 10°C, making it at that instant an equivalent to Bath 1's starting condition. From this point on, Bath 2 will cool the exact same way that Bath 1 did from 10°C to 0°C. Since it took a non-zero amount of time for Bath 2 to cool from 20°C to 10°C, we now have our answer to the question. Hot water cools at a faster rate than cold water, but will take a longer total time to cool to some lower temperature.

With regard to your question about molecules growing with temperature, your second thought is more correct than your first. A molecule "heats up" when you give the system thermal energy (increase the temperature), and this in turn gives the molecule more kinetic energy (the movement you were referring to). We have to be careful not to think of this as the size of the molecule itself increasing, but rather that the molecule is starting to vibrate and take up more space. These vibrations of the molecules cause them to spread out, increasing the volume and causing the overall system of molecules (aka the matter you can actually see) to grow.

This can be seen in many examples: for instance this is why if you've ever left a plastic water bottle in a hot car, you'll notice when you come back the bottle feels as if it's about to burst. All of the water molecules inside are moving faster than they were before, increasing the volume of the water, and therefore increasing the pressure on the bottle.