How does it work if I cast Abandon while having an empowered Mel, Newly Awakened but they Rebuttal my spell?

In the case of Rebuttal, gaining control of a spell is not the same thing as countering that spell. Effects that prevent spells being countered don’t prevent their controller from changing. If Rebuttal is played choosing your spell, and they pay [A] during resolution of Rebuttal, they will gain control of it. However, them not paying [A] does not mean the spell will be countered.