The derived category of an abelian category has a t-structure, so obviously that's the first thing you want.
To a t-structure corresponds a heart, which is an abelian category, whose derived category might be different than the one you started with.
To further complicate matters, as you noted, you could have a heart whose derived category *is* actually equivalent to the one you started with, but the heart itself is not the original abelian category.

It's not clear what you can recover from a triangulated category alone, or even from a triangulated category + t-structure.
If you like algebraic geometry and are willing to consider additional structures then you can recover the abelian category. The derived category of a scheme, considered as a monoidal category (coming from tensor products) recovers the original scheme (and thus the abelian category).
The same is true if you start with a variety with an ample canonical bundle, then the category plus the bundle do recover the variety.

Somehow this flexibility of derived categories is a nice feature, as it gives rise to interesting (and hidden?) "symmetries" and "relationships" between spaces.

As per the second question I can only think of what Sasha said, that is stability conditions.
Given a stability condition one automatically gets a heart of a t-structure (which again may not have anything to do with the original abelian category) and the slices of the stability condition may be seen as a continuous family of t-structures.
It would indeed be really nice to have such a thing as a moduli space of t-structures!