Is there a correlation between the different types of weapons and the particular opponent in a scene? Basically, is there a benefit to using a certain type of weapon over another when fighting against, say, Klingons?...or Romulans?...or Orions? And is there a list somewhere for which type of weapon is most effective against which opponent in STO?

Nope, STO doesn't do this. Your weapon choice is all about what the weapons do instead of who you're fighting. This could be just an extension of the other freedoms built into the system, or it could be because of the old skill system railroading you into picking a single energy type.