I always thought that Angier, like Bordon, cloned himself just once, and then, for his 100 performances, again like Bordon, used that one clone for the show. I did not think that he cloned himself every time, because if he did, there would be 101 Angiers at the end of the shows. So when Angier drowned, I also did not think about him setting up Borden, I actually never thought about that. I just thought that he, the real Angier, had set up the show so that there would only be one Angier in the end (not just end of the show, but end like during his retirement). We also do not know how many shows there were out of the 100 before Angier died, so did Angier want to go through with all of his shows, or just some, until he could get Bordon? I thought that he would be the one appearing at the prestige, and his clone would drown. If he used just one clone every time he performed, did he use the machine every time? What machine would he have used? If he didn't and used the machine every time to clone himself every time, how did he know that it would work every time? Like Tesla said, "An exact science is not an exact science." How did he know it wouldn't stop working on him like it did for Tesla? But some things are made clear when I think about what Angier said, "Every night, I did not know if I would be the one in the box, or the one at the prestige."
Quite a few questions but here it goes:
The movie clearly shows 1 Angiers always drowning. He used the machine each time he performed the show to clone himself and then killed the Angiers that stayed on the stage to prevent multiple Angiers from walking around at the same time. This is part of his vanity as he wanted to get all credit. He could have cloned himself once and done something like Bordon but that's not in his nature. In the last scenes of the movie you can also see the past dead clones in their tanks at the storage.
As for the revenge part, framing Bordon for murder was clearly part of the plan all along. he still hates him for the death of his wife and is very keen on being the best magician. The 2 go to extremes to get the credit.
Considering the exact science remark: I don't think that was a comment on the fact that it might stop working one day. It had to do with the progress and functionality of the machine at that point in time.
When Cutter first goes to the address on the back of the card and sees it's Robert, Robert makes a comment that the show has to be huge because Bordon 'must come to see it' and try to figure it out; in other words, he's already planning to set Bordon up for his 'murder'. The original machine does not have a trap door; Angiers has installed that for 3 reasons; primarily to kill the original (the clone is created away from the machine), but also to provide a gaff that allows a bit of doubt to enter the audience's mind, and finally to steer Bordon downstairs thinking he'll catch Angier's secret again; don't forget that's how Angiers broke his leg, when Bordon removed the cushioning, and then rode up the other side to spoil Angiers' reputation. Angiers is standing there when Bordon not only goes onstage lightly disguised, but walks into the machine and stomps on the floor looking for traps.
There is also the bird in the cage metaphor to be considered, that of getting your hands dirty to do good magic, like the spring box that kills the canary, though the magician produces a second one pretending it's the same bird; there's a murder in the trick, since Angiers has decided he can't be in two places at the same time for whatever reason (me, I'd like to be able to do that). A "person" is drowning each night even though his clone is being recreated, and it's the original Angiers who's dying and knows he's walking to his death each time. That's why the stage hands are blind, Cutter's not allowed, and the box (covered) is quietly taken from the performance house to the derelict stage each night; there's a dead body in every box though the blind stagehands don't know it, created while waiting to lure Bordon, and you see at least 8, probably more, surrounding the two men at the end when the surviving Bordon shoots Angiers/lord whosis. The very last shot of the film is a pan to one of the cadavers, still in his drowning box, off to the left.
The machine itself never needed a trap door; recall that the hat never seemed to leave, and the cat just walked away; it wasn't until Robert was in the woods that he saw there had been dozens of hats and at least one other cat replicated; nobody until then realized the machine was doing something. However, instead of teleportation, it was a replicator. So the film makers are suggesting that this machine actually worked, was not a stage prop, and the magician (being who he was) shaped its capabilities into a magic trick (as opposed to, say, a doctor maybe could clone a spare body for a rich person to have as parts in a medical thriller, or a Star Wars villain could clone an army).
The part I didn't understand (lots going on in the end) was how Cutter came to have custody of the child to turn her over to Bordon, and what the knowing head-nods were between the two of them, unless I assume an unseen alliance formed between Bordon and Cutter after Cutter found out Angiers was still alive, maybe out of Cutter's guilt at having testified against Bordon when Angiers hadn't really died. There was a certain betrayal involved in that, since Cutter was the eyewitness to Bordon being under the stage, and made the assumption that Bordon had sabotaged the drowning box lock to not open; however, Cutter was also stooged into providing that testimony by not knowing about all the previous murders. But that still doesn't explain how the child was with him when Bordon came for her; he was doing magic tricks for her from the beginning of the film, amusing her while they were waiting.
As to how the clone knew to appear or not; the front apron of the stage was so flimsy, when they were under it in previous scenes you could see the stage lighting through many cracks between the boards. The audience would have been able to hear Bordon screaming for help (Bordon was down there before the trap opened and started screaming immediately_) and the banging of the axe on the glass while he was trying to break Angiers out. That would be enough noise to alert the clone that he/they had finally trapped Bordon and not to come out; no further cues to the clone would be needed. The previous nights, he waited long enough to NOT hear any ruckus and knew to complete the Prestige, but I suppose the ultimate Prestige was presenting the trick in a way that got his rival hung for a murder he didn't commit.