The space of marked groups can be used to show some properties of surface groups. For example, see the following question: Is the center of the fundamental group of the double torus trivial?
I found also some interesting results in the appendix of Model Theory by Hodges.
Consider the following problem: If $A,B,C$ are three groups such that $A \times C \simeq B \times C$; when $A$ and $B$ are isomorphic? For example:
Theorem: Let $G$ and $H$ be finitely generated finite-by-nilpotent groups. Then $G \times \mathbb{Z} \simeq H \times \mathbb{Z}$ iff $G$ and $H$ are elementary equivalent.
See Cancellation and Elementary Equivalence of Finitely Generated Finite-By-Nilpotent Groups or Cancellation of abelian groups of finite rank modulo elementary equivalence by Francis Oger.
Shelah solved the following problems using model theory:
Theorem: Any uncountable group $G$ of cardinality $\lambda$ has at least $\lambda$ subgroups not conjugate in pairs.
See Uncountable groups have many nonconjugate subgroups.
Theorem: There is an uncountable group whose proper subgroups are all countable.
See On a problem of Kurosh, Johnson groups, and applications.
In On some conjectures connected with complete sentences, Makowski relates the problem "Is there an infinite finitely presented group with a finite number of conjugacy classes?" to the existence of a specific kind of theory (theorem 2.6).
A group $G$ is said linear of degree $n$ if it is embedable into $GL(n,F)$ for some field $F$. In Barwise's book, Handbook of mathematical logic, there is a proof of
Theorem: Let $G$ be a group. If every finitely-generated subgroup of $G$ is linear of degree $n$, then $G$ is linear of degree $n$.