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?
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.
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.