Baxter is turned into a wild-haired, comedic Mad Scientist in the first cartoon (you weren’t allowed to have black villains in children’s cartoons back then), acting as a nerdy loser working for Shredder, and later one of the show’s many stock “Mutant Supervillains”. He appeared in the Archie continuity in the same vein, but the writers quickly ignored him- he is arrested by the police and then never seen again. They go a different route with villains and thus Baxter never becomes “The Fly”. The 2003 series makes him more like the Mirage version, but ramps up how crazy he is. The 2000s & Nickelodeon cartoon and the original Mirage comics got more play out of him as an elite scientist (and all made him actually black… while keeping April a Caucasian redhead), but he eventually went nuts in them anyhow. In the 2000s one, there’s a sick running gag where he is mutilated further and further with each failure, while the Nick toon combined the two previous cartoons’ Baxter into making him a goofy fly-man, but kept the fact that he was black.
In Rise of the TMNT, he is “Baxter Stockboy”, a young black kid who is a genius working (for no pay) in the store his parents own, and begs for online subscribers and such. In the 2014 movies, he gets a cameo in the first, but appears as Shredder’s minion in the second, playing by Tyler Perry- he annoys the boss and is sent away before the finale. He was apparently going to be a mutant fly in the next one, but they cancelled the movies instead.
In most continuities, he is the creator of the “Mousers”- robotic minions.
Baxter Stockman was a recurring foe for the Turtles in the 1980s cartoon, starting out as a goofy, nerdy scientist under The Shredder’s employ. This low-rent Jeff Goldblum (all of the nerd, but none of the sexy), he was intially just some hapless loser- an innocent scientist conned by the Shredder into using his rodent-exterminating technology for EVIL- his Mousers being the perfect method to getting rid of Shredder’s old nemesis, Splinter. Baxter was a Mirage original, but the show’s writers wisely turned the black character white, as having the show’s ONLY BLACK CHARACTER being a bad guy (one frequently abused and bossed-around by another) was deemed a bad idea. Baxter was thrown in an insane asylum at the end of his story arc (his talk of the giant talking rat and turtles was seen as ridiculous), and returned as a much more vengeful and evil lunatic.
To add “More Mutants” to the Merch-driven show, Baxter soon transformed and became an even BIGGER example of the Butt-Monkey by turning into a Human Fly (horrifyingly, Krang decided to DISINTEGRATE him because he no longer needed the scientist around- a fly buzzed in alongside him, leading to the transformation). And most of his appearances ended with him in exile or being stranded somewhere. Pretty much the universe’s victim by this point (a tragic origin story, since he was actually a fairly mild-mannered dude), he had a bunch of science-based weird stories about transformations and new inventions and junk, but he eventually just turned into a joke. There’s a weird bit where he is exiled in space in Season 4, returns with no explanation in Season 5, then pops up in Season 7 still in exile as if the intermediate appearance never happened.
Frequently appearing as an early foe because that’s what they were in the Mirage Comics, Mousers fall away as the various TMNT shows age- in many continuities, they appear only once. But still, they’re an iconic bit of business: an unending army of Robot Mooks that do nothing but eat whatever’s in their path. stuff. They were invented by Baxter Stockman to rid the city of rats- this brings them into conflict with Splinter. Their toy was rather prominent, as it came out quite early- nonetheless, it is WAY out of scale with the TV show appearance, where they were about the size of a turkey. The toy (with a wind-up walking feature) is more or less scaled like a Bullmastiff.