Abandon

What happens when you play Abandon on a spell that was played for its Flow cost?

It will be banished when it attempts to leave the chain.

"[T]hen banish it" in flow's text is short for "if it would leave the chain after becoming a finalized chain item, and leaving the chain wasn't instructed by its own execution, banish it instead." Abandon returns the countered card to hand after countering it, but for the purposes of flow's "then banish it", this is also leaving the chain. No matter how it happens, as long as the flow spell leaving the chain wasn't instructed by the execution of the spell itself, it will be banished.

This also applies to effects like Kai'Sa, Evolutionary from Origins, or Fizz, Trickster from Spiritforged.

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.