So I am doing an experiment with gravity car, basically dropping a hanging mass and spins the axle using a pulley system. 
And then I thought although the work formula W=Fd doesn't prove that gravity does work on the car, the kinetic energy comes from the change in gravitational potential energy. How is the energy transferred if gravity doesn't do work on the car? What is the relationship between energy transfer and work done? Do they contradict each other or what? Thanks!
[Physics] How does a gravity car work (and where does the energy come from)
newtonian-gravitynewtonian-mechanicspotential energy
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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.
There's always confusion with this topic when it's not well explained. It's all inside "work-energy theorem", which says
$$\Delta E_k = W$$
But we'll make a distinction here: work done by conservative forces and work done by non conservative forces:
$$ \Delta E_k = W_C + W_{NC} $$
And now, we just call "minus potential energy" to the work done by conservative ones
$$W_{C}:= -\Delta E_p$$
We do this for convenience. We can do it, because a conservative force is such taht can be written as a substraction of a certain function $B$ like this:
$$W_C=B(\vec{x_f})-B(\vec{x_0}) $$
We just decide to define $E_p=-B$, so $W_{C}=-\Delta E_p$. We include that minus sign so that we can take it to the LHS:
$$ \Delta E_k = W_C + W_{NC} $$ $$ \Delta E_k = -\Delta E_p + W_{NC} $$ $$ \Delta E_k + \Delta E_p = W_{NC} $$ $$ \Delta E_m = W_{NC} $$
So the increment in mechanical energy is always equal to the work done by non-conservative forces. If there are no non-conservative forces, then $\Delta E_m=0$ and energy is conserved (that's why we call them like that.
(read it slowly and understand it well)
So, having this in mind, I think your confusion arises because of that famous "artificial" negative sign.
There are many formulas, and it's typicall to have a mess. It's all about surnames: $\Delta E_k = W_{Total}$, but $\Delta E_m=W_{NC}$. The subindices are the key.
The force of engines is non-conservative. Hence, their work contributes to total mechanical energy.
Gravity is conservative, so we can work with its potential energy.
If there is no increase of kinetic energy, that means
$0 + \Delta E_p = W_{NC}$
So engines are only increasing potential energy. But that means
$$-W_C = W_{NC}$$
Of course, if there's no gain in KE, no acceleration, there's equilibrium. The work of the engines is compensating the work of gravity.
- Negative work is always positive $\Delta E_p$, by definition.
- More altitude means more $E_p$, you are right. But here energy is not conserved (engines). Normally, increasing height would decrease $E_k$, but we're adding work so taht $E_k$ stays constant.
- $\Delta E_k=0$ implies $W_{Total}=0$. That means gravity is making negative work, and engines are doing positive work (equilibrium). The thing is that potential energy variation is minus gravity's work.
Best Answer
Gravity does work on the falling block, which then does work on the car (through some gearing). We could detail it a bit more and say that the block actually does work on the wheels that are being turned, which then in turn cause the friction from the ground to do work on the car.
The work done by gravity is what we call change in potential energy.