I have a rigid body response dilema as to their real world behaviour.

Consider a cube (in space- no gravity) that moves its direction and hits a static (super stable) platform orthogonaly like this:

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I then wonder wheather the fact that the cube mass center being over the static platform in the direction of cube movement, will couse the cube to not to "try" hang over the edge at all, or, the fact that some part of mass is still over the edge to move freely, will couse the cube to emit some tendence of movement to the right over the egde. So will it slightly bend over the platform or not at all?

And second question is what would happen if there was attractive force in the direction of cube movement against the platform?

In reality, neither object can be perfectly stationary or have infinite mass, so some slight rotation and movement along the X axis is going to occur. Also, collisions are continuous over a short period of time and varying amounts of forces are applied as the collision progressively alters the objects.

The result depends a lot on what the masses of the two objects are. If the cube is much more massive, the platform will move down and slightly to the left while rotating clockwise, while the cube will continue down and to the right and also spin clockwise.

If the platform (and whatever it's attached to) is much more massive, the cube may bounce up and slightly to the right while spinning slightly clockwise, and the platform will deform elastically and return to its approximate original position relative to the large massive body it's attached to.

Depending on surface friction, the cube may slide off the right side of the platform.

Nothing in reality is perfectly stiff, either. Depending on the stiffness of the cube (imagine Jello), it may deform and "bend" over the edge.

If the platform (and whatever it's attached to) is much more massive, the cube may bounce up and slightly to the right while spinning slightly clockwise

I think this is a very beautifull funded answer. Would this apply for the modification of: perfectly stiff objects, and the platform not able to move and absorb no energy (meaning platform will couse nearly perpetum mobile return of reaction energy back to the cube, that is undeformable too, and will transform entire reaction energy to kinetcs). Would cube, then, move just perfectly up, or still would have emited a clockwise rotation and translation to the right?

Your illustration above wasn't clear to me, so, if your cube is traveling to the left, then it would rebound to the right with a counter-clockwise rotation.

The cube travels down, orthogonaly onto the platform.

Yes, I have "eliminated" certain real-world conditions, moving it to more theoretical level, to spectate rather only few certain real-world behaviorals of rigid bodies. Those conditions though can happen (omitting even heat on contact) to such level of derivation that one could spectate "only" that certain behavioral- wheather the part of mass with free tendence would couse the cube to recieve tendence of kinetics to the right, or "not at all" (as for the derivation acceptance of those scaled up "perfect" conditions).

Would this apply for the modification of: perfectly stiff objects, and the platform not able to move and absorb no energy (meaning platform will couse nearly perpetum mobile return of reaction energy back to the cube, that is undeformable too, and will transform entire reaction energy to kinetcs). Would cube, then, move just perfectly up, or still would have emited a clockwise rotation and translation to the right?

As long as the collision physics still use some approximation of http://en.wikipedia.org/wiki/Coulomb%27s_law , then I believe it would still have a clockwise rotation, but the bounce angle would probably have less magnitude in the rightward direction.

... use some approximation of [ Coulomb's law ...

Coulomb's Law has to do with electrical charge. I don't see how that relates to ideal rigid body collision. Can you explain?

The OP seems to be asking about what would happen if you had a physics engine that was a mix between rigid body dynamics and real-world subatomic-force-based physics, so I'm trying to mentally simulate what would happen under those conditions.