Rebuttal

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.

What new choices can I make when I play Rebuttal and pay [A]?

You can choose any number of targets or modes to make new choices for. The rest remain the same.

If you are remaking choices in this way, any choices you choose not to remake must be legal choices. You can't choose to have some but not all choices be legal choices. Of course, you can always choose not to make new choices. In that case, any of the current choices will remain in their current state, whether that state is illegal or not.

If a spell has some rules or requirements about the choices made for it, those apply regardless of who controls that spell. For example, if a player plays a spell from face down, they must target game objects located at the battlefield the spell was played from unless the instructions of that spell indicate that a target must be chosen from some other location. Gaining control of such a spell doesn't change this requirement when making new choices for the spell.

We'll look at shaping up this section in a future rules update.

I have a Ravenbloom Student on the board and I haven't played any spells this turn. What happens when I play Rebuttal targeting my opponent's Stupefy and pay [A] on resolution? After everything resolves, how much will Battering Ram cost? Will my opponent's Legion abilities be active?

Ravenbloom Student will trigger twice, Battering Ram will cost [4], and your opponent's Legion abilities will be active.

Ravenbloom Student's triggered ability checks whether a spell that you control resolves. You gained control of the Stupefy, and it resolved, so you will have triggered Ravenbloom Student's ability. Battering Ram and Legion, on the other hand, check whether cards were finalized. Although you gained control of Stupefy, you did not finalize it; you got it after it was already finalized by your opponent. Thus, your opponent's Legion abilities will be active, and Battering Ram will only receive a [1] Energy discount.

Relevant Rules: 419.4., 419.4.a., 419.4.b.