Mel, Newly Awakened

What happens when I play Moonfall choosing an enemy unit while I control an empowered Mel, Newly Awakened?

The chosen unit will be given -3 [M]. All other enemy units at the chosen battlefield will be given -2 [M].

Mel, Newly Awakened's empowered ability is a replacement effect. It replaces the event of a spell or ability you control giving a unit it chooses -[M]. The event it replaces that with is the spell or ability giving an additional -1 [M].

Although Moonfall gives -[M] to multiple units, only one of them is chosen, and thus the only event that is eligible to be replaced is the chosen unit being given -2 [M]. It will be replaced with that unit being given -3 [M]. The events of the other units being given -[M] will not be affected by this replacement effect.

Relevant Rules: 370.1., 370.1.b.

What about casting Stupefy while I have an empowered Mel, Newly Awakened?

The chosen unit will be given -2 [M], to a minimum of 1 [M]. If it was 2 [M], it will become 1 [M].

In the case of Stupefy, Mel, Newly Awakened replaces the event of Stupefy giving -1 [M] with it giving -2 [M] instead. However, a replacement effect that replaces an event modified by the game effect that generated that event will inherit any of those modifications. In this case, Stupefy modifies the "give -1 [M]" game action and thus event with "to a minimum of 1 [M]." The replaced event will inherit the same modification.

Relevant Rules: 375.

Speaking of Mel, can my opponent play Abandon on my spell while I control an empowered Mel, Newly Awakened? What happens?

They can play Abandon targeting your spell. On resolution, they will Predict, but nothing else will happen.

Mel, Newly Awakened says "your spells and abilities can't be countered." If the spell can't be countered, Abandon will attempt to counter it and fail, and then it will replace the event of it being put in the Trash with it being returned to its owner's hand. However, that event never occurred, so there is nothing for the replacement effect to replace. Thus, Abandon will resolve to no effect. The replacement effect only functions within the limit of the resolution of the ability (it's not a delayed replacement effect that specifies a duration), so even if the spell resolves later, it will go to the Trash as normal.

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.

Note that this logic also applies to Mel, Newly Awakened and Switcheroo.