The first thing I want to explain is what I believe to be the central, most important journal note that you find in the game. It's the one in which Mandus is explaining to his kids how the Aztecs would cut out the hearts of sacrificial humans in order to keep the sky from falling. What would happen if the sky fell? It'd be the end of the world. It'd be absolute chaos and destruction.
Where was Mandus when he had this discussion with his children? In the very temple where he found the Orb. This would imply that - much as Mandus describes the Orb whispering to him - it was whispering to the Aztecs as well, telling them to commit all these sacrifices to prevent the chaos and destruction that was to come. Keep this in mind.
Now, onto the questions mentioned in the review, first up is "What's the deal with all the pigs?" This is quite obviously a euphemism for people, in fact towards the end of the game it even says this. We're all pigs. The whole world is A Machine for Pigs, made only to be slaughtered. A key component to this game - and one that is necessary in order to understand the game's story - is that Mandus is quite delightfully mad. Mad in the "mad scientist" kind of way. He listened to the Orb and was granted visions of the future. In it he saw the absolute filth that was humanity, he saw how we're all disgusting pigs. He saw the horrors that man will commit upon itself, "chaos and destruction" as I mentioned before, and decided that he was going to do something about it. So he creates a grand machine for slaughtering humans and pigs alike. The left-overs are ground up and sold as a cheap food supply (as seen in one of the loading screen pictures that shows a bunch of meat flowing out of a big pipe into a vat that's being used to fill up little carts being hauled away by workers). "Why pigs?" Because much like how Batman uses the image of a bat to signify the terror he wishes to inflict upon criminals, Mandus uses pigs to signify the disgusting abominations that he sees mankind to be.
Other possible explanations - this one actually coming from Myth Busters which apparently makes it "known" science
- is that apparently pigs and humans have remarkably similar physiologies. If you're planning on building an army of human-animal hybrids, it'd make sense to start with an animal that is already physiologically similar to humans. Though personally I think using pigs makes sense purely on the fact that Mandus sees all of humanity as nothing but a bunch of pigs. Add to it the fact that pigs are a major food source - and being able to provide a cheap food source is integral to Mandus' plan - also goes into it. This brings us to "What's the deal with the church?"
The Church is explained in a couple notes you actually find inside the church itself. Essentially Mandus was giving out a very cheap food source to the poor, orphans, and other down-trodden folk of the city. This caused masses of people to gravitate towards his charity. All of those people would be rounded up and turned into pig-men. It's all about how perfectly efficient the machine is. Throughout the game are references to how every single aspect of the machine is used to run another aspect of the machine. Heat from one part of the machine is taken away from a place that needs to be cool and it is instead used to heat another part of the machine. Noxious gasses from all the left-overs are turned into a fuel source. The flow of all the entrails and waste water is used to power turbines. The same is true here: the meaty waste of creating human-pig hybrids is used to lure in more humans so that they can be turned into human-pig hybrids, which makes more food to lure in more people, etc.
Now onto the "why". As in: "Why create a vast army of pig-men?" Again: the visions of the future that Mandus was granted. His goal wasn't to create a charitable food supply, his goal was only ever to create an army of man-pigs to unleash upon society. The dead would be harvested and taken to The Machine, turned into more man-pigs, and sent out to kill some more people and continue the process. He was going to wipe clean the scum of the earth by killing everyone on it, thus preventing the "chaos and destruction" that he saw in his visions. Again, tying back into the note about the Aztec sacrifices, one of the lines in it was "They were certainly not savages, it was likely that they just couldn't sacrifice enough to prevent the sky from falling." With the modern technology of the day, Mandus COULD sacrifice enough to "prevent the sky from falling". Really I see similarities between this story and Mass Effect 3. Just like how The Reapers were supposed to wipe out advanced organic life before it became too advanced and created synthetic life that would wipe out ALL organic life, so too would Mandus wipe out all human life and replace it with pig-men so that way there could be no WWI, no WWII, no Atomic Bomb, none of the horrors that he had seen in his visions from the Orb.
That's right, Mandus was a Reaper.
That's why all of this was necessary. The Orb drove Mandus to madness. It caused him to kill his children in the temple, rip out their hearts, and sacrifice them to it. Why did it pick Mandus? Because he was a wealthy industrialist who could complete the designs that it had for the human race. Again, the Aztecs just couldn't sacrifice enough to prevent the sky from falling...but Mandus had the means to do so. Why did he sacrifice his children to it? Well at one point one of the kids says "You can hear the sea", but that's not really "whispering". It does, however, imply that the Orb was giving off some kind of "signal" (for lack of a better term). I'm pretty sure that once you realize that the voice in your head is the voice of the Machine - and by extension, the Orb itself - it mentions that in order to become partners with it you first had to bath it in the blood of your children. You had to sacrifice them in order to give the Orb its voice. From there, the Orb became "The Engineer" that Mandus mentions partnered with him. It was someone who shared his vision of the future because it was the very source of his vision of the future.
As for how Mandus ends up with amnesia, that's actually explained in another journal note. Mandus is pondering over what makes a man, is he defined by his actions? Can a man's very soul be remade like clockwork? He talks about how he's going to reach for the exposed wires. He hopes that he'll wake up and be able to start anew, but if he dies, then maybe it's for the best that he dies amongst his creations. This was obviously written after Mandus began sabotaging his own machine, and also explains why - if killing himself would forever power-down the machine - he didn't just kill himself in the first place. Well he was hoping that if he zapped himself with enough electricity he'd lose his memory and just start over with a completely fresh new life. He had already manually shut down the machine, so he was hoping he could start over. Clearly this was not the case as when he woke up, the madness was still there and it guided him through the steps necessary to restart the machine. That does, however, bring us to the one question that I don't have the answer for, but rather a shaky theory on.
To me, the big question is "How was Mandus connected to his machine?" I have to say that I think Torque's interpretation was correct: the chair you sit in with all the metal arms and such does seem to imply that it was designed specifically to kill Mandus. In fact the little monologue at the end definitely implies that Mandus was now dead. Buuuuuuut how does killing himself stop the machine? If he was completely mad while listening to the stone, why would he build in such a failsafe? I don't know if Torque is right about the Orb being inside Mandus the whole time...everything I found suggested that Mandus just took it home and put it on the mantel. But seeing as how his explanation makes about as much sense as anything I could come up with, I'll go with it. Perhaps that machine at the end was the very thing that put the orb inside Mandus to being with and all it did at the end was take the Orb back out. We already know through notes that Mandus was bringing dead things back to life, so maybe he is, himself, a "Compound X" zombie.
Even so, that STILL doesn't answer how Mandus dying - or even the Orb being removed from his chest if that is indeed the case - would forever stop the machine from working again. I gotta say I don't buy into the "there were two Mandus-es" as Torque suggested, there really isn't anything to suggest that's the case. But it's obvious that the machine was somehow connected to Mandus in some way. I'll probably have to play through the game another time or two to see if there's any way to answer that question. As for the rest of the questions, I think the answers are indeed there and not particularly vague or hard to piece together. Hopefully my explanations have shown that the story, at least, wasn't as incoherent and piece-meal as you first suspected.