Hi @Beyondforce,
I don't really know if these lower-level aspects of the joint solve are documented anywhere. Maybe they should be, I agree.
Regarding:
1. Right now I'm very confused whether it matters the order you choose the components or not!
The answer is: It only matters during the command preview animation, that is when you see the Animate control in the dialog:

Once the command has been OK'd, the order doesn't matter any more. If both components are free to move, the solver can move whichever its algorithms pick. To control this, the best way is to "lock down" the one you don't want to move. That makes it unambiguous.
Regarding:
2. The "solver" is not following the basic principles of the Fusion 360 Joints!
Joints are allowing freedom of movements. In my example, the freedom of rotation should ONLY apply to the Gear and Not the Pin.
In order to create a joint, you need 2 components, but it doesn't mean that the joint should be apply to both of them.
The Fusion joint solver is what we internally call a pure "variational" solver. Meaning that there are no directional relationships. A revolute joint is not "from one component to another", it is an equal relationship: Those two components can rotate with respect to each other. A good analogy is to think of the sketch solver (which, by the way, in Fusion is the same solver). If you have two lines with a parallel constraint between them, either line can move, and the other will follow. You wouldn't want the sketch solver to restrict you to only move the "first" line, and have the "second" line react, and not be able to move the second line.
I would not say that I am 100% confident that there is nothing wrong with the Revolute joint, or that what we looked at here is what is going on in the other case you are looking at. However, I am probably 95% sure that in this case, what I said is the reason for the behavior you see. If you just ground those pins, I'll bet you get the behavior you want.
This is a good discussion. This is not obvious stuff, and as you say, it is not well documented, and further, not obvious at first.
Jeff
Jeff Strater
Engineering Director