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!