Two-Face being mutated into a (two-faced) cat may sound a bit arbitrary. But then it dawns: it fits because cats are fickle! For instance, some cats act indecisive depending on whether or not they want you to scratch them. Or they can show affection to someone one instant and a cold shoulder the next. What better than a cat to represent how arbitrary Two-Face and his coin-flipping decisions are?
Alternatively, Two-Face is a cat because he's Schrödinger's cat. Appropriate, as he's both Batman's enemy and ally, depending on the flip of a coin.
Additionally, cats in real life are extremely rarely born with two faces, a condition called diprosopus, with the afflicted cats referred to as Janus cats. This not only provides a plausible result of the mutagen, but also fits in perfectly with Two-Face's theme of duality.
For that matter, Joker's a cobra instead of something more obvious like a hyena. Why? He tells us earlier—cobras tend to be venomous...
Of course Batman won the fight against all four Turtles. In most continuities, the Shredder is capable of beating all four of them simultaneously and in this version is a slightly-above-even match for Batman. If Batman is on Shredder's level, then it stands to reason his fight with the Turtles would go the same way.
Of course the TMNT would barge in on Batman's business. Gotham is based off New York City, so they are actually fellow city folk.
Gotham isn't terribly far from New York either, as most Batman media places it on the coast of the New Jersey side of the Delaware Bay - generally a 2-3 hour commute by car. As a hot spot for crime of the street, organized, and super varieties, it isn't too far fetched for Shredder to look into resource acquisition when it's so close by.
The two universes be in the same setting rather than some alternate dimensional shenanigans raises quite a few implications...
The mutated Mook who was a T-Rex was cured by having Robin throw multiple syringes into his body though his mouth. So when his body shrunk upon being cured...
