Heating and cooling are rarely nicely split up in time. While you're heating something it is also cooling and while you're cooling something it is also warming up. This usually limits your ability to heat something up or to cool it because at some point these two are in balance.
No. The outer layer of the hot thing will quickly heat up to the temperature of the hot medium, then it will cool down as is stays in contact with the middle layer, which will heat up itself. The same thing repeats in layers all over the object.
No, why do you think the outer layer will cool down? There will be heat transfer between the outer layer and the layer within, but at no point in time will the temperature of the outer layer be cooler than it was at an earlier point in time, so it will never cool down.
But the thing which is "dirtier" is the wider universe, where the energy you used to do that sorting has higher entropy.
At the scale of atoms and molecules in a gas, you can also sort them into high-energy particles on one side of a barrier and low-energy particles on the other side, and now there's a heat difference you can run a heat engine. This is totally a thing you can do with the right devices — but those devices will necessarily consume more energy than you get from a heat engine running on that heat difference.
That’s because the box isn’t a closed system, you’ve interacted with in and you have spent energy sorting the bricks. It’s the same story with humans on earth, things get sorted because of the energy inout from the Sun.
At the boundary between water and air, molecules are constantly snapping back and forth between being liquid and vapor. It's just that an equilibrium has been reached that makes the system appear static.
Molecules don't have innate solid or liquid properties, but they do operate as a solid or liquid, correct? It has nothing to do with them having a temperature of their own, and everything to do with how they are currently acting with their peers. IIUC.
You can call the limitations of your knowledge about the exact state of a molecule a temperature, but that doesn't mean that any single molecule has an actual physical temperature in the same way as a collection of particles. You can say that a passing car had a speed between 60 mph and 70 mph because you couldn't measure it more exact, but the car has at any given moment an exact speed independent of your knowledge of it, not a distribution of speeds.
That is exactly GPS point: a car doesn't have a distribution of speeds unless you measure it relative to something else whereas a liquid or a solid does have a temperature, regardless of anything else. Depending on what the something else is that you measure your car's speed by you may decide it goes forward, backward or is motionless.
When you start talking about individual molecules, atoms or particles the whole concept of temperature becomes very counterintuitive. Think of it as a substitute for motion or vibration if you wish and even that is grossly inaccurate (but less so...).
Maybe this will help: a gas in a container has a pressure and a density as well as a temperature, all of which are properties of the gas and not of the individual gas molecules. A single molecule that the gas is made up from does not have a density, it doesn't have a temperature and it doesn't have a pressure. What it may have though is a speed relative to something else, and when it hits the something else it may impart some energy relative to that speed difference.
