The backstabber's denunciation cancels your DOF, so you are free to smack him into next Tuesday. Denounce him back and then DOW him when your troops are in position. To avoid potential genocidal maniac warmonger diplo penalties, consider leaving him with one inconsequential, ice-bound city to occupy as he considers the error of his ways (the diplo penalty for eliminating a civ is not automatic, but not worth risking either).
As for what to do with his cities, when you conquer them do a quick assessment of their strategic value. If they have some luxes (even duplicates are OK, since you can sell/trade them), decent buildings left after the sacking (I've gotten some that still had universities, coliseums and such) or have other strategic merit (blocking position, etc.), they are probably worth keeping.
If it's not worth keeping, immediately raze it. If it's worth keeping, puppet it. Once the citizens cease revolting (1 turn per surviving pop), the city will go on gold focus and will eventually provide good amounts of gold and, to a lesser extent, culture and science. TP as many of the food tiles as you can to limit growth and preserve happiness. If it's a capital city and looks like it would be productive city, you can annex it (after the citizens cease rebelling and you can rush-buy a courthouse with gold), but annexing will cause your policy costs to rise (as with any other non-puppet city).
EDIT: One other thought on his last city; requires the existence of one of your allied CSs next to his last city and the timing can be tricky, but you can beat that last city's health down to 0 and then let your allied CS deliver the coup de grâce (only done it once in my games, but it was fun). No civ-elimination diplo penalty for you, even though you would have orchestrated it
