1) You won't get an "official response" from Paizo on this.
2) You get both attacks, one from cleaving finish (and knocking an opponent to 0 or less HP), and one from cleave for hitting your first opponent.
You can try showing your GM this thread once you get more responses confirming what I've stated here. If your GM is open minded, then they should be convinced. If not.... find a new GM IMO. Close minded GM's that aren't willing to learn more about the game and how it works aren't really worth playing with.
1) To expand on what bbangerter said, Paizo developers do not have time to resolve every player/GM dispute that gets brought to the messageboards. If you are very, very lucky a developer may weigh in with their personal (non-official) opinion in their spare time.
2) If you didn't get to continue your Cleave then Cleaving Finish would be more of a liability than a benefit. Feats are supposed to help you significantly, not hinder you. See what your GM thinks of that argument.
3) Showing the GM this thread is fine, but I think bbangerter is going a bit far on ditching the GM if they aren't convinced by the opinions of a bunch of people they don't know.
Yes Cleaving finish can trigger off of cleave. As for official the best you will get quickly is the consensus of the forum.
It's not clear to me whether you get the second Cleave attack first or the Cleaving Finish attack. I guess I'd leave that to the player.
Maybe your GM worries that Cleave would be too powerful, especially with something similar on top. It can look quite impressive, especially at the first levels, but it has serious limitations:
a) You have to hit the first foe - which is not a given, to the very least because of the risk of rolling a 1.
b) Damaging multiple opponents at once doesn't weaken them until they are killed.
c) Your enemies have to be in proper position - respective you have to move to get into position.
Cleaving Finish will effectively add another attack every few rounds - you need 2+ enemies within range and one of them must have so little HP (left / from the beginning) to kill it. So from my gut feeling both feats roughly have the same power level: Good but not gamebreaking.
As a GM, I'd let a PC with Cleave etc. have their fun against stupid enemies (e.g. zombies), but vs. smart foes the player will have to invest some effort to make it work (e.g. retreat into a corner and hope that they will follow).