Let $\mathcal{M}$ be an infinite model of a first-order language, and for each $n$, let $\mathcal{B}\_n$ be the algebra of definable sets of $n$-tuples from $|\mathcal{M}|$.


1. Given $\{\mathcal{B}\_n\mid \_{n\in\mathbb{N}}\}$ (and, obviously, $|\mathcal{M}|$, is it possible to describe explicitly some $\mathcal{M}'$ whose definable sets are $\{\mathcal{B}\_n\}$? (I think the question makes the most sense assuming that the language itself is unknown, so I'm asking if there's a natural way to invent a language and a model which gives the chosen definable sets. Obviously there is such a model, namely the original one, but I'm open minded about what it would mean to construct $\mathcal{M}'$ "explicitly".)
2. How well do the definable sets "pin down" the model? Need two models with the same definable sets in the same language be elementarily equivalent? Can anything be said about the relationship between two different models of different languages, but with the same definable sets?
3. There are some obvious restrictions on the $\mathcal{B}\_n$: $\mathcal{B}\_n\times\mathcal{B}\_m\subseteq\mathcal{B}\_{n+ m}$, each $\mathcal{B}\_n$ is a Boolean algebra, $\mathcal{B}\_m$ contains all projections from $\mathcal{B}\_{m+n}$. Are there any others?



Reconstructing a model from its definable sets
The answer to the first part of #2 is no. In a language with (just) one binary relation, let the two models be $\omega$ with the relation interpreted as $<$ in one model and as $>$ in the other.


For #1, the canonical choice would be to take all the relations from the $\mathcal{B}\_n$'s as the interpretations of relation symbols. Presumably, you want a smaller language, but I don't see any good criterion for what would be small enough.


For #3, I think the restrictions you list are all you need (provided "Boolean algebra" means subalgebra of the power set of $|\mathcal{M}|^n$ for the appropriate $n$). They seem to be exactly what you need to do quantifier elimination for the language and interpretation I suggested for #1. 


