ct.category theory - What is the relationship between t-structure and Torsion pair?

The two notions are related in the sense that they share a common generalization, namely the notion of torsion pair on a pre-triangulated category (this term has at least two meanings, here we mean a category which has compatible left and right triangulations - it covers several cases including triangulated categories and quasi-abelian categories). The reference for this material is

A. Beligiannis and I. Reiten: ''Homological and Homotopical Aspects of Torsion Theories''

which is available from Beligiannis' homepage. In fact one can take the analogy further and consider the analogy between TTF-triples on an abelian category and recollement of triangulated categories.

There is also another connection given by tilting theory. Suppose that $(mathcal{T},mathcal{F})$ is a torsion pair on an abelian category $mathbf{A}$. Then we can obtain a t-structure on $D= D^b(mathbf{A})$ by setting
$D^{leq 0} = { Xin D ; vert ; H^i(X)=0 ; text{for} ; i>0, H^0(X)in mathcal{T} }$
and
$D^{geq 0} = { Xin D ; vert ; H^i(X)=0 ; text{for} ; i<-1, H^{-1}(X)in mathcal{F} }$
For more information on this (in particular for some characterizations of when taking the derived category of the heart obtain from this t-structure is equivalent to $D$) one can see "Tilting in Abelian categories and quasitilted algebras" By Dieter Happel, Idun Reiten, Sverre O. Smalø.

I hope that at least goes some of the way toward answering (1) and (2).

As far as (3) is concerned I am not completely sure what to say. Certainly one can reconstruct a quasi-compact quasi-separated scheme from its derived category using the tensor structure, and if the scheme is particularly nice one can use the Serre functor. I am not aware of (or have forgotten if I knew) a way of reconstructing a scheme via t-structures (I guess one can use strictly localizaing subcategories which are particularly nice t-structures or take the heart of the standard one). One certainly can't just look at all t-structures - even for $D(mathbb{Z})$ there is a proper class of t-structures.
In the abelian case the closest thing I can think of is taking the spectrum of indecomposable injectives. This is not directly torsion theoretic but it is true that injectives control hereditary torsion theories in the sense that every hereditary torsion theory in a Grothendieck abelian category has as its torsion class the left orthogonal to some injective object.

A particularly nice special case when one can really make the connection precise is the following (due to Krause). Suppose that $mathbf{A}$ is a locally coherent Grothendieck abelian category i.e., it is a Grothendieck abelian category with a generating set of finitely presented objects and the finitely presented objects form an abelian subcategory. Then one can topologize the spectrum of indecomposable injectives in such a way that there is a bijection between hereditary torsion theories of finite type (those for which the right adjoint to the inclusion also commutes with filtered colimits) and closed subsets of the spectrum.

One last thought for the moment - although one can think of t-structures and torsion theories on abelian categories as common specializations of one more general definition the analogy can be misleading. However, there is a reasonably good analogy between hereditary torsion theories of finite type and smashing subcategories which can be made precise (again this is due to Krause). The heart of this is that every smashing subcategory of a compactly generated triangulated category is generated by an ideal of maps between compact objects. Corresponding to such an ideal there is a hereditary torsion theory of finite type in the category of additive presheaves of abelian groups on the compact objects. Something you may find particularly interesting about this (I certainly do) is that it links the theory of smashing subcategories (and the telescope conjecture) to the spectrum of indecomposable injectives in a nice abelian category.