There are only two split brushes. A and B are part of the same brush. They are bent out of shape. Before you re-assemble the motor, bend the metal brushes carefully until A and B are close together (like the two parts of C)and then adjust them to make A+B and C symmetrical with each other and nearly touching at the tips.
It is going to be difficult to get the back on the motor without bending the brushes out of shape again. The easiest way would be to push the shaft far enough out that the brushes can be aligned on the commutator, before moving the back and shaft into position together. To do that, you need access to the drive end of the motor shaft. What is connected to it? Does it have a gear on the shaft? Is there enough distance between it and the motor body that the shaft can be pushed in to give enough clearance at the commutator end to get the brushes re-aligned?

It is not clear to me whether the two brushes (A and B) of the same brush should touch or not.

As for the other questions you asked, I have uploaded the photos in post #12 ... if they are not enough I will upload more, I am waiting for your reply.

Thank you!

 

I measured it.

In both cases (from A,B to the red wire and from C to the black wire) with the multimeter set to "200 scale" resistance, I measured a resistance that varies from 4ohm to 15ohm ... it fluctuated a lot because the hand shakes and the brushes are not fixed but bend at the slightest touch with the tip

Well, that would indicate the problem is not between the input and the brushes. Did you measure the continuity of the windings as I outlined earlier?

 

Well, that would indicate the problem is not between the input and the brushes. Did you measure the continuity of the windings as I outlined earlier?

I would say to proceed with photos as in the case of the brushes :/
1) What I see are 2 permanent magnets in the stator and windings in the rotor that are 120 degrees out of phase with each other.

If you want, point me in this top photo where to put the multimeter tips.


2) Regardless, I am thinking of powering it directly (like in the picture below) and seeing if it runs.
The problem is that I only have a portable power supply at home that provides 5V and 5A DC. Probably too much? Will I burning something?
(this is a photo taken online, but of course it is the exact same DC motor)

 

It will not run like that. It needs to be assembled with the armature between the two magnets, and both bearings supporting it so that it doesn't touch the magnets. I think the picture is illustrating how to measure the armature resistance.
There are three copper segments on the commutator. measure the resistance of each with respect to its adjacent segment. The measurements should show a fairly low resistance and all should be similar. That will prove that the armature is not damaged.
The motor probably stopped running because the brushes no longer made good contact with the commutator. If the armature is good, try re-assembling it, making sure that the brushes are bent to make good contact. A voltage higher than the original 3V will burn the simple metal brushes and the motor will stop working again.