The Sorgenfrey line, or the uncountable product of the two-point discrete space, are both zero dimensional and yet satisfy D(X)=c. So this operation does not behave very nicely with non metrizable spaces.

How do you define the associated "non standard space" to a standard space?

It's probably a bit late, but David MJC is simply... wrong. For a non compact space, it is not sufficient that the map strictly reduces distances, as can be seen by the map x + 1/x which has no fixed points. The computation with geometric series is essential - the mapping has to be a contraction.

There's a natural bijection between the set of continuous functions from a $T_0$ space to the Sierpinski space and the topology of the space, but there's no such bijection between the topology of a regular space and the the set of continuous functions to the unit interval. I don't know category theory so I don't know how to make this precise but I think you should get the intuitive feeling that the Sierpinski space is in some sense canonical and minimal. Anyway the answer given is what I've been looking for (I actually read the book but managed to miss that part)

I don't think, because it still makes a reference to an external object.