Mew-EX

If I use Sudowoodo's "Watch and Learn" attack on the turn after my opponent's Mew EX used its "Versatile" Ability, can "Watch and Learn" copy the attack that Mew EX used?

Yes, that works.

Since a BREAK Pokemon "retains the attacks, Abilities, Weakness, Resistance, and Retreat Cost of its previous Evolution" can a Mew EX with the "Versatile" Ability use an attack of the Pokemon that the BREAK Pokemon evolved from (providing it meets energy requirements, etc.)?

Yes, it can. The underlying Pokemon's attacks are part of the BREAK Evolved Pokemon and are therefore able to be copied by Versatile.

What happens if Mew EX uses its "Versatile" ability to copy Xerneas EX's "X Blast" attack? Can it use "X Blast" again the following turn or not?

When Mew EX attacks, Versatile uses the attack name specified. So when it attacks with X Blast, Mew EX cannot use that attack twice in a row.

For Mew EX's "Versatile" Ability, do I need to have the same energy types as the attack I'm copying, or will any type of energy work as long as I have the correct number of energies?

You must meet the energy requirements as to both type and number for the attack Mew EX is copying. So if the attack requires two Water energy, you will need to have at least two Water energy attached to Mew EX.

If my Wailord uses "Giant Wave" this turn, can my opponent's Mew EX's "Versatile" Ability use that attack on the turn following mine?

Yes. The effect of "this Pokemon can't use Giant Wave next turn" applies only to the Pokemon that attacked with it.