There's no conflict in storylines... They fit together perfectly. They did slip up on the writing in Homeworld 2 and state that the Kushan core was, in fact, the original - But that's it. The Kushan core was reverse-engineered and recreated to the exact specification of the damaged core on Kharak. (Which, some believe, actually holds some pieces and modules of the original core which was destroyed in the Exodus.)
"How'd they manage to get that technological knowhow to do so? I mean if they could reverse-engineer it, how come other empires did not?"
The other empires had no core. The Kushan did. It stands to reason that the Bentusi's core is not reverse engineered because, as far as we know, there has never been a battle where it was notable damaged. - Until the Dust Wars. (The war in Homeworld 2.)
The Kushan core is only damaged because the 'Empire's' (And, by empire, I mean the ancient Hiigaran empire.) flagship, the Sajuuk's-Wrath, hyperspaced into the Angel Moon. It's almost certain that such an impact was damage the core beyond reason. (This is probably why, on the level, you can see the Angel Moon has an abundance of cracks and craters. Though some believe them to just be lights of cities.)
The Kushan core may have also decayed, in it's damaged state, during it's slumber under the deserts sands and damaged during the landing (And, possible crash.) of the Khar-Toba.
To answer your question: If you have a technology, and you have an abundance of time, you can always learn how to recreate it and use it. (Although, in this case, why do they not build multiple cores? It is believed that fully recreated cores do not function and that the new Kushan core only does because they used parts from the original core that were not damaged which are vital for it's function.)