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 Borden 'must come to see it' and try to figure it out; in other words, he's already planning to set Borden 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 Borden downstairs thinking he'll catch Angier's secret again; don't forget that's how Angiers broke his leg, when Borden removed the cushioning, and then rode up the other side to spoil Angiers' reputation. Angiers is standing there when Borden 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 Borden, and you see at least 8, probably more, surrounding the two men at the end when the surviving Borden 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 Borden, and what the knowing head-nods were between the two of them, unless I assume an unseen alliance formed between Borden and Cutter after Cutter found out Angiers was still alive, maybe out of Cutter's guilt at having testified against Borden when Angiers hadn't really died. There was a certain betrayal involved in that, since Cutter was the eyewitness to Borden being under the stage, and made the assumption that Borden 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 Borden 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 Borden screaming for help (Borden 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 Borden 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.