But, adaptation can happen, but only within limits, within the kind of the organism. And, you can get variation, but also, within limits, within the kind of the organism.
What sets these "limits?"
What is a "kind?"
This is awfully well-trodden territory here at EvC, but I've yet to see even a good try at answers to those two questions. You might want to read some old threads here in Biological Evolution before you offer your answers.
I cannot define kind exactly, it would more likely be like a family....
That doesn't seem very likely biblically, as Leviticus 11 (or Deut.14) has "kinds" of herons, hawks, and ravens, and lists owls, great owls, and little owls separately, apparently as "kinds" of their own.
and only the stronger and the most fitting organism would promote itself,
That's "the organism better adapted to the current environment" that "promotes itself," or, rather, leaves offspring. Horseshoe crabs aren't known for their ferocity or cunning, but they survive and reproduce very nicely in their little niche. Tigers are famously fierce and strong, but as we take away their natural habitat, they'll all die out. "Strong" doesn't mean "fit" in biology.
If the theory of natural selection is true, wouldn't we be able to survive another "big bang"?
I presume you mean "mass extinction?" The last big one of those, at the end of the Cretaceous, killed off all the Big Strong Dinosaurs and left a bunch of rat-sized mammals. Big and Strong was the wrong thing to be 64,000,000 years ago, at least on land. The changed environment didn't leave big guys up on the surface enough food or shelter, apparently. If the mass extinction we humans are causing leads to enough depletion of our fellow organisms, we may go that way, too: the cockroaches, rats, and bacteria could take over this time.