I will discuss several kinds of connections between the areas mentioned in the title. A common theme is the structure of group objects in various categories.
1. Firstly, model-theoretic methods associated with the theory of
stable groups lead to the following solution to questions of Kolchin.
Proposition (Pillay). If $G$ is a connected differential
algebraic group then $G$ can be differential rationally embedded in an
algebraic group. Moreover $G$ has a unique maximal normal connected
linear differential alegbraic subgroup, and the quotient group is
embeddable in an abelian variety.
2. Manin proved in his work on the Mordell conjecture for function fields that any abelian variety (in char. 0) admits a differential rational embedding into some $G_a\times\cdot\times G_a$. Applying this to an elliptic curve with differential transcendental $j$-invariant, one can prove Proposition (Hrushovski-Sokolovic). There are continuum many countable differentially closed fields of characteristic 0.
3. Let $A$ be a commutative algebraic group over an algebraically
closed field $F$, and $\Gamma$ an arbitrary subgroup of $A$. Let us
say that $(F,A,\Gamma)$ is of Lang-type if for every $n>\omega$ and
any subvariety X of $A^n=A\times\cdots\times A$, $X\cap \Gamma^n$ is
a finite union of cosets of subgroups of $A^n$. Lang conjectured (and
Faltings recently proved) that $(F,A,\Gamma)$ is of Lang-type whenever
${\rm char}(F)=0$, $A$ is a semiabelian variety, and $\Gamma$ is a
finite rank subgroup of $A$. Being of Lang-type is actually a
model-theoretic property, namely
Remark. $(K,A,\Gamma)$ is of Lang-type iff ${\rm
Th}(K,+,\cdots,\Gamma)$ is stable and the predicate $\Gamma$ is
1-based.
Using the (Buium)-Manin homomorphism mentioned above, an analogue
in characteristic $p$ and a dichotomy theorem for certain strongly
minimal sets, Hrushovski has given a model-theoretic proof of the
function field version of Lang's conjecture, in all characteristics, a
special case of which is:
Proposition. Let $k>F$ be algebraically closed fields.
Let $A$ be a semiabelian variety over $F$ with trivial $k$-trace.
Let $\Gamma$ be a finite rank subgroup of $A$. Then $(F,A,\Gamma)$
is of Lang-type.
Similar ideas give rise to a model-theoretic proof of the Manin-Mumford Conjecture.