I incorporated the TMNT as NPCs in my Street Fighter: The Storytelling Game campaign many years ago, and it worked out rather well.
Animal-Mutations as seen in the cartoon
The Street Fighter Players' Guide had rules for animal hybrids, although most folks considered them overpowered. I don't think that'd be much of an issue if the entire team and/or most of the villains were hybrids or balanced around them.
Martial Arts Capabilities & Quick combat
Street Fighter was all about martial arts (and to a lesser extent, melee combat) in a fairly cinematic style, although it did try very hard to emulate the special maneuvers, etc. from the video game. There are rules for "fast and loose" combat if you don't want to use the actual system, though.
Eh. It's kinda comic-booky but I ran a fairly gritty campaign. That's in your hands, really. There are no rules at all for character death (characters merely get KO'd at zero health with any spillover causing aggravated damage), so the lethality of the campaign is entirely up to you.
Oriental feel (Samurai honour, etc)
Same. This a setting detail, not something rules should handle. although the game does rank Honor and Glory as something fighters aspire to gain and Honor affects Willpower and Chi recovery.
Some sort of Rule of Cool stunt system
If you use the Combat Cards system (I didn't), one of the cards was called "Stunt". There's a section in the rules on how to handle them, but basically, yes; do something cool and see if it works. You're not hamstrung with a high difficulty roll if you want to do something out of the ordinary.
Some mysticism or magic accessible to the players
Fireballs, teleporting, ninja magic, elemental powers... the whole Focus-based group of Special Maneuvers falls under this.
The ability to use High-levels of technology
The Players' Guide also contains rules for cyborgs which aren't bad. Otherwise, it's really easy to "hack in" blaster rifles and robots and such; I modeled them after existing weapons and raised the damage slightly, and just treated the robots as NPCs. The average Street Fighter was capable of punching brick anyway, so why not?
Oh, and one of the styles in the Players' Guide is Ninjitsu. So, there you go.