The modern physics has no problem with that. It's just apply extra energy to rotation part with postulating law of translational momentum conservation. However, it's say nothing about where is another opposite angular momentum.
That's what classical physics does too. Except the part about another opposite angular momentum. They both say the whole system (both rods) have a net zero angular momentum.
1. Rotational and translational motion is standalone natural phenomenon because it may be initiated from one event and it should have it's own law of momentum conservation.
I'm sure you can formulate it so they're a single concept. Actually when I was in school I always imagined rotational motion to be a special case of translational motion that they just used because it made things simpler.
Suppose you had a massless rod with a point mass on each end. If it's rotating you could consider it as having angular momentum. Alternatively you could just use the translational momentum of the point masses. It'd get a bit complicated because they're changing direction as it rotates, but that's OK, the rod provides a force which causes them to change direction.
2. What will happen if there repulsed objects will never collide with other objects? The energy will divided asymmetrically between two sides.
Yes, but so what? Who says they both have to have the same energy?
