In the situation sketched in the question, only point 1. and 2. are relevant, as there is not any significant heat transfer between the bodies.
What coefficients should be in the first law according to the above assumptions:
There is no single/unique answer to this - you can choose the coefficients as you like.
The 1st law of thermodynamics says that
$$d U=Q-W$$
which is the same as you describe ($U$ is internal energy, $Q$ is heat, and $W$ is work). But, I could just as well have written it as
$$d U=Q+W$$
or with signs added in other ways. It depends on what I mean by $Q$ and $W$ in the specific case. To choose for example the version $d U=Q+W$, I have to state at the same time that $Q$ is heat absorbed and $W$ is work done on the body (by an external force). If I used $d U=Q-W$, then $W$ would have been work done by the system.
The method to remember it: Consider energy positive when it is entering (or absorbed by or added to) a body or system, and consider energy negative when it is leaving. Choosing this sign convention makes energy simple to add. If I lift a book up to a shelf, then I do work on the book (+W) by adding potential energy to the system. If I push a box across the floor, then I do work on it (+W) by adding kinetic energy to it. If on the other hand, the heavy box comes sliding towards me and hits me, so I am pushed away, then the box did work on me ($-W$). The work the box did on me is energy lost from itself; the amount of kinetic energy remaining in the box will necessarily be smaller after the collision.
Edit: I guess the second law might introduce a≤ (or is it ≥?) instead of an equal sign, since I talk about entropy.
The usual equation for entropy of a system is $S\geq\int\frac{dQ}{T}$. It states that entropy can never decrease in a process, if the whole isolated system is considered.
From comments
Work done ON A SYSTEM or BY A SYSTEM, I have never, ever seen a definition of. Work done by a force I have seen defined, on the other hand
You are correct, that only forces do work. The statement, "work done by a system" simply means that the system applies a force, which is doing work (on something else). In your meteor example, the meteor as the system is doing work on the planet by pulling in it toward itself through its gravitational force. (Note: The work done by the meteor on the planet is very, very small, since the displacement of the planet will be very, very small).
My guess is: work done by a system, is the thermodynamical force generated by the system multiplied by its conjugate variable on the surroundings. As in my example here: a gravitational forcefield is generated by the system on the surroundings, and the conjugate variable displacement on the surroundings is multiplied by it.
Yes, this is the mathematical definition, in short written as:
$$W=\vec F \cdot \vec s$$
or in general for non-constant forces:
$$W=\int \vec F \cdot d\vec s$$
where $\vec s$ is the displacement vector.
Don't be surprised that physics has a lot of definitions that are circular. Ultimately, we are just describing the universe.
Work and energy have been defined in a certain way in newtonian physics to explain a kinematic model of reality. This is a model, not reality - you will find no such thing in reality. However, in many scenarios, it is close enough to reality to be useful.
For example, let's say that a human has a 10% efficiency at converting food to mechanical work. So if you spend 1000 kJ of food energy to press against a wall, are you doing 1000 kJ of work, or 100 kJ of work, or 0 kJ of work?
In strict mechanical sense, you did no work whatsoever, and all of the energy you used was wasted as heat. If you instead used this energy to push a locomotive, you would have wasted "only" 900 kJ of the energy as heat, with 100 kJ being work. But the locomotive has its own friction, and it wil stop eventually, wasting all the energy as heat again. And overall, you did expend all those 1000 kJ of food energy that is never coming back.
All of those are simplifications. Kinematics is concerned with things that move. Using models is all about understanding the limits of such models. You're trying to explain thermodynamics using kinematics - this is actually quite possible (e.g. the kinematic theory of heat), but not quite as simple as you make it. Let's look at the fire example. You say there is no displacement, and therefore no work. Now, within the usual context kinematics is used, you are entirely correct - all of that energy is wasted, and you should have used it to drive a piston or something to change it to useful work.
Make a clear note here: what is useful work is entirely a human concept - it's all 100% relevant only within the context of your goals; if you used that "waste" to heat your house, it would have been useful work. It so happens that if you look closer, you'll see that the heat from the fire does produce movement. Individual molecules forming the wood wiggle more and more, some of them breaking free and reforming, and rising with the hot air away from the fire, while also drawing in colder air from the surroundings to feed the fire further. There's a lot of displacement - individual molecules accelerate and slow down, move and bounce around... But make no mistake, the fact that kinematics can satisfactorily explain a huge part of thermodynamics is just a bonus - nobody claimed that kinematics explains 100% of the universe. It was a model to explain how macroscopic objects move in everyday scenarios. It didn't try to explain fire.
For your specific questions, you really shouldn't ask multiple questions in one question. It gets very messy. But to address them quickly:
- There is no potential energy in the kinematic model. The concept is defined for bound states, which do not really exist as a concept in kinematics. In other models, you might see that there's a difference between, say, potential energy and kinetic energy - no such thing really exists in reality. You need to understand the context of the model.
- In a perfectly kinematic world, this is 100% correct. However, as noted before, kinematics isn't a 100% accurate description of reality, and there are other considerations that apply, such as the fact that humans have limited work rate, limited ability to apply force, and the materials we are built of aren't infinitely tough, perfectly inflexible and don't exist in perfect isolation from all the outside (and inside) effects. In real world applications of models, these differences are usually eliminated through understanding the limits of given models, and using various "fixup" constants - and if that isn't good enough, picking (or making) a better model.
- You're mixing up many different models at different levels of abstraction and of different scope so confusion is inevitable. Within the simplified context of kinetics, there is no concept of "potential energy". You simply have energy that can be used to do work, and that's it; it doesn't care about how that energy is used to do work, about the efficiency of doing so etc. In another context, it might be very useful to think of energy and mass as being the same thing - and in yet another, they might be considered interchangeable at a certain ratio, or perhaps in a certain direction, or at a certain rate. It's all about what you're trying to do.
- How is that equation useful? That's the only thing that matters about both definitions and equations. I can define a million things that are completely useless if I wanted to - but what's the point?
- Within the original context, those aren't considered at all. Within a more realistic context, both heat and sound are also kinematic.
The reason you have so much trouble finding the answer to your questions on physics sites and forums is that the question doesn't make much sense in physics. It's more about the philosophy of science, and the idea of building models of the world that try to describe reality to an approximation that happens to be useful to us. You think that those words have an inherent meaning that is applicable in any possible context - this simply isn't true. From the very inception of the idea of physics, people have known that it isn't (and never will be) an accurate representation of reality; and we've known for a very long time that, for example, different observers may disagree on the energy of one object. You just need to understand where a given model is useful, and pick the right model for the job. Don't try to drive a screw with a garden rake.
Best Answer
Work is the physical concept that we use to relate force and energy. It's good to have a clear idea about 'work'.
When you push a box on a table, you give a speed to the box. So your force caused the increase of kinetic energy of the box. So your force has done a positive work on the box. If you try to stop the moving box with an opposite force, your force is slowing down the box, which means sucking the kinetic energy of the box. Therefore, your force has done a negative work on the box.
But if you try to push a wall, you can't move that wall. Therefore it is called that no energy is transferred. That means no work is done. This obeys the mathematical formula $W=FS$.
As a gist, work in mechanics is the relationship between force and energy.
Boiling water is related to thermodynamics. In that subject work is directly difined as
Thus it is easy to comprehend 'work' as a measure of energy transferred, in any subject regardless mechanics or thermodynamics or whatever.