I've been checking the difference between the five words in question and I admit it's not yet entirely clear me how to do distinguish between them, so I'd be glad for any further insights (if there is anything more to be said about them). Here's what I've found thus far:

To REFUTE means 'to say or show conclusively to be false or illogical' and applies primarily to assertions or arguments;
To CONFUTE means 'to prove wrong' and applies primarily to the person or agency making the assertion or argument; refute decisively
to REBUT a statement is to offer clear evidence or a reasoned argument against it; to argue the falseness of
to IMPUGN is to dispute the truth, validity, or honesty of, to cast doubt upon by criticizing, to attack as false or questionable;
to OPPUGN is to dispute the truth or validity of, to strive against, oppose, or call into question.
All are quite formal. 'Confute' is a rarity that few would understand. I've never even heard 'oppugn'.