In the attached video I rotate the cube with my mouse wheel. It seems to be doing exactly what you depicted in the sketch.
What’s stopping you from adjusting the axis value in the CombineRotators node in the 2nd screenshot I posted?
Because the pyramide (or the cube in your video) must follow the cursor.
My pyramide is already well placed when I click. It is attached to the surface of the sphere, and I can move my mouse without displace it anymore. But impossible to align the front of the pyramide with my cursor, while maintaining the position according to the sphere 
Because the pyramide (or the cube in
your video) must follow the cursor.
It is following the cursor. You want it to sit on the surface of the sphere, right?
It is attached to the surface of the
sphere, and I can move my mouse
without displace it anymore.
Stop updating its location in Tick. It will stay in place.
But impossible to align the front of
the pyramide with my cursor, while
maintaining the position according to
the sphere
Attach the pyramid to the sphere, it will become its child and move with it.
From what I gather, you want to attach the pyramid to the sphere, rotate it and then be able to just move the sphere while the pyramid stays attached to the sphere. Please do correct me if I misunderstood you.
I have 2 steps :
1/ I click on a button to get a pyramid. This pyramid follow my cursor, and is placed on the surface according to center of the sphere. I can place my pyramide where I want.
2/ I click to place my pyramid. It keep its rotation from before, and it is well placed on the surface. Nothing more is updated every tick. So far, so good.
I don’t release my click, and I can move my mouse without the pyramide follow it. But now I need my pyramid front watching to my mouse.
My English level does not allow me to express myself more clearly x)
Here is a video of one of my test (with a 2tan cacul for a virtual circle around pyramid… but no success) :
To rotate it toward the cursor, you can use Find Look at Rotation node. If you want to have it flipped on the side, combine rotators again.
Yes, this is the first thing I did. But this only work at the top of the sphere. If I put my pyramid at the bottom, for example, the base of the pyramide is on the surface, and the head is in the sphere… The pyramid ignore the previous rotation and keep the X, Y, Z basic plan.
EDIT : I saw your video after my answer to your post! Your cube is out of the sphere. What I was looking for is rotate localy the pyramid, always on the sphere, and according to the center of the sphere (+ according to mouse location), exactly lik in my video (the surface is a sphere!).
You should then look into InverseTransform[rot/dir] nodes, it will allow you to create a rotation in relative space.