My superhero is, of course, a cat (since cats are obviously super)! He* doesn't have a name; his kind don't communicate using language (preferring telepathy, of course); when humans talk about him they generally use common nouns (e.g., "that cat over there", or "that cute little kitten I always see") or pronouns (e.g., "she", or "he"). She looks like an ordinary cat and does not have any distinguishing characteristics (he's white with either blue-green or purple eyes, depending on some lighting phenomenon/which color is handy for the guy drawing the comics; they're officially "bluish").
She mainly saves people/animals from evil villains using his power of extreme cuteness/mind control. She can also be in two places at once and teleport short distances, provided no one is looking. He also has extremely good reflexes (including the ability to fall from thirty-story buildings without getting hurt). Oh, and he can die and not die at the same time, which she does at one point at the hands of her arch-enemy, Schrödinger. (Note that he does not wish to be called "Schrödinger's cat".)
He also has an extreme obsession with calculus, and before doing anything he often insists on figuring out every possible motion equation that's involved using integrals and derivatives. She's sometimes considered lazy, but he explains that by saying that it's "easier to integrate a constant acceleration of zero twice to find my position** than to integrate the constantly changing acceleration of me walking."
* or she; her/his gender is a quantum superposition of "female" and "male", and has not yet been observed. The pronouns "she" and "he" can be used interchangeably for this superhero, and it would not be incorrect for both to be used in the same sentence.
** Integrate once: v = v0. Integrate again: d = d0 + v0t.
Major superpowers: He can be in two places or have two opposite qualities at the same time (until those qualities are observed, of course...); he can read minds, control minds (to an extent), and communicate telepathically; he can teleport under certain circumstances; he can resurrect himself (nine times only); he has extremely good reflexes (including the ability to fall from thirty-story buildings without getting hurt). Oh, and he can die and not die at the same time.
Costume: None, but not very many people recognize him. She avoids being well-known
How he became super: Genetic predisposition, plus events explained in the origin story
Why she does heroic things: Because cats are, by nature, good.
What he eats for breakfast: How should I know?
How he fights: Mind control. Occasionally uses force when necessary; he's generally a pacifist.
Physical laws she breaks: All Newtonian mechanics laws, plus the fact that he can influence and detect what happens in someone else's mind without any physical connection or energy transfer (see first item on this list).
Catch phrase: meow
Other things she does: Eat, sleep, repeat.
Where he lives: It is impossible for me to be certain about that, but I do know that she currently has a momentum of exactly 5 kg·m/s.
When they make a movie about your hero, who do they cast in the lead role? A computer generated image. (Any dialog would, of course, use speech synthesis, a technology that has existed since before the invention of the Macintosh, yet people still insist on using actors for voices in movies...)
The following stories take place in the universe that the cat lives in: