While this may be a stretch, it's possible Splinter manufactured them with raw materials he found in the sewer.
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" but how they are still usable is kinda odd since Donatello's brand of MacGyvering probably shouldn't include weaponsmiting.
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.
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.