My guess would be that he found them, perhaps not in the sewer/storm-drain/streets/whatever, but at a local dump. It's not hard to purchase such things, and crazier stuff has been "just thrown away" . How they are still usable is kinda odd, since Donatello's brand of MacGyvering probably shouldn't include weaponsmithing.
Raiding Foot safe houses for supplies? Splinter apparently heard their plans as a rat, and they were monitoring the Foot in the present.
He ordered them online? Remember this is set in 2014. They could all have online jobs to get money and you can order combat ready martial arts weapons if you know where to look. P.O. box address at an isolated location where they could disable the security cameras. This day and age it'd be easy.
Can somebody please explain whatever wacky Technobabble the Foot used to instantly find the Turtles?
Tracker in the business card, I thought.
Yes, there was a tracker in the business card Sacks gave April. There's a blink-and-you'll-miss-it moment where she whips it out, as Splinter explains that Sacks is in league with the Foot, right before a cut to the Foot actually using magic surveillance technology to localize them.
Why in the name of Krang didn't April show Bernadette the photo she had of the Turtles jumping across the rooftops!?
I assume she intended to, and her initial conversation with Bernadette was simply to introduce the general concept of giant goddamn Turtles and then provide solid evidence with the picture (you can see April started off with the basic, describing who the Turtles were, then evolved to more solid stuff like the repeated markings and such), but Bernadette cut her off mid-way through and fired her. She did make note that she hadn't slept, her mind probably wasn't at top form.
Even if she did show it to her, it might not have made much difference. The picture could've been photoshopped, for all she knows.
Also it's a really shitty picture that doesn't really show much.
Why were the Turtles attacking Shredder one-at-a-time during the big fight? I thought they were supposed to be a team, not common villain mooks.
That may be the point. The Turtles being disorganized and not very well united in battle is a running theme (what with Raphael and Leo's bickering), and this is evidenced by them attacking separately (and subsequently get their asses kicked), but then they decide to fight together and united as one (with things like the "buck buck" tactic and the human rope to save April), and it is only then they achieve success against the Shredder.
Generally it's agreed that to make both Shredder and Splinter look more awesome, the four turtles must always need all four of them working as a team to beat the Shredder.
Who flushes a martial arts book down into the sewer?
It could have slipped from someone's hands near a storm drain. I've heard of people losing things like that.
Shouldn't Splinter have mutagen too? I mean, the Foot show no interest in capturing him, and there's a line about how the mutagen can "save" him, but...HE'S A GIANT NINJA RAT! DID HE GET THAT WAY THROUGH SOME OTHER MEANS!?!
The Turtles are more reliable due to their being four, they can be sure of the same compound. Also as they're younger, they can survive the process longer, and therefore get more mutagen.
It seems a safe bet that he did, given the transformations and their past at the lab. As to why he wasn't a target, remember that they only knew about the turtles, since April only told Sacks about them, not knowing about Splinter being alive at that point. There's also the fact that Splinter was going one-on-one with The Shredder, so he may have been considered too dangerous to capture; following that, he was left for dead either way, just like Raph was when they took the other Turtles.
Added adrenaline is a dangerous thing anyway. Considering the amount they're given under the hideously low blood pressure the turtles should have had three heart attacks in a matter of minutes. You'd think Donny, being a genius, would know that?
The movie makes extra effort to show us that the people on the ground are affected by the battle between the turtles and Shredder on top of Sacks' tower - people are screaming and dodging falling debris and the police are pulling out assault rifles and such as if it's an armed bank robbery. But why on Earth did no one, not even a news reporting agency (like, I don't know, Channel 6 News), think of sending a helicopter up to see what's going on, when clearly some serious stuff is hitting the fan? This isn't the Avengers, where flying spaceships would shoot down aircraft in the area, so there doesn't seem to be a reason why no one can fly a helicopter over to see what's going on. Did Sacks pay off all of the news agencies agencies or something? I know it would ruin the 'turtles have to stay anonymous' plot point, but is there a reasonable in-universe explanation for mass news media complacency?
Everything takes time. Choppers can't be there instantly — and police often won't call the media until they know what's going on. Unless someone is in the know, or a chopper's already in the air, it can take most of an hour to get coverage.
How exactly did Vern know it was April calling when she was doing it from a payphone in the sewer? He very obviously looked at the phone and grinned a bit, like he was seeing her name come up from his contacts, so it couldn't have been something like a collect call.
There's caller ID, and then there's experience. Vern and April have a history together as colleagues, and given April's tendency to snoop around just about anywhere, he probably guessed (correctly) that she was calling him from a payphone.
How did Sacks expect people not to notice that the poison was coming from his skyscraper? Not to mention, what exactly what the Shredder going to get out of it? "Power," he said, but how does poisoning people get you power over them?
The poison may actually be odorless and colorless (we see a red gas coming from the tower just before the machine shuts down, but it's possible that this is just red smoke used to test the disbursement system prior to activation). If this is the case, then the poison could be disbursed continuously for a time without anyone being the wiser.
As for power... well, having the cure for an outbreak before anyone else does can give you power over those who are sick and dying of it. People are willing to give up money and more in the short term if their lives depend on it. It also means that not only can they name the price of the cure, but they can also name who can get the cure and who can't. I can see Sacks and Shredder working together to give the real cure to the Foot Clan's allies / the rich / corrupt politicians willing to give out political favors, while giving a lower-grade version of the cure (or a placebo) to those who pay less / enemies of the Foot Clan. In short, it would give you power to decide who lives and who dies, and a little extra pocket money for Sacks Industries.
The scenes where the Shredder is speaking Japanese seem pretty pointless. It's established that he can speak perfect English and yet, he feels the need to speak in his native tongue. What is the point of having that in the movie if Karai and Sacks don't speak it as well?
I remember Karai being able to speak Japanese, and Sachs should be able to, though I don't remember them having a single conversation in Japanese. As for your question, Rule of Cool?
Why is young April allowed to feed Splinter and the Turtles pizza? Why is she even allowed into the lab, not to mention handling the lab animals? If you are testing anything, wouldn't it make a lot more sense to have the animals you are testing it on on a controlled diet? It's like Sacks and Dr. O'Neill just threw the scientific method out the window...