LOL! Even when she's getting shot repeatedly with a gun, and keeps on coming at you like Jason Voorhees or Michael Myers?
I thought the exact same thing when you see Batman getting shot in the Batman 89 rooftop scene, that part was creepy and looked out of a horror flick.
She probably had a Kevlar under her suit.
That's probably it, it's actually lifted off of a 1946 Batman comic.
Selina was brought back from death by cats. Why would cats chewing her fingers and running all over her revive her, unless they somehow used some kind of their life force or something like that to bring her back.
The licking woke her up from unconsciousness.
One thing happens a lot in films, a dog licks the asleep owner and he wakes up due to the licking.
She also suddenly became remarkably agile and a brilliant fighter without any training.
Well, the film is called Batman Returns not Catwoman Begins, they can't dwell on every single detail since they have to focus on other things as well not just one character that's not the protagonist, so she probably took training but they never show that in detail.
Or maybe, she was a agile person that had a accident then had amnesia, then become secretary and then another accident via her boss pushing her out the window that led her back to have the brain of the agile person.

I don't mean to say that literally but just a guess considering the character's origin in the film is inspired by the amnesiac air-hostess origin from Batman #62 (1950).
Right. But I cannot deny her getting shot FOUR times at close range and just walking it off like it was nothing.
In Batman #35 (1946) "Nine Lives Has the Cat", Catwoman goes to hire new thugs but they say no because they're not impressed with the fact that Batman has defeated her many times nor are they scared/take her seriously then she thinks of something to scare them with due to knowing that Batman makes a impact on thugs through intimidation (it's never shown how she prepares the trick) but on the next day, the annoyed thug shoots at her (when she comes to see the thugs again) but she doesn't die and she claims that she has nine lives.
It's pretty obvious that Kevlar armor was used in that issue.
If you're not convinced then I can post the two pages of this issue.
She was pale white, no breath was seen coming from her mouth in the cold, like it was just when she hit the ground etc.
What do you think happened to her?
There was just a cut/bruise on the forehead, no sign of broken body and no pool of blood.
As for, pale white??? Bruce Wayne looked pale white when the Bat-Signal light lit up on his face in the Wayne Manor study scene where he just broods, it's all about lighting or Tim Burton's deliberate visual style.