Can the map sending a presentation to its group be considered as a functor?
Yes, yes, yes, to all three questions.
And it can be done very generally and very nicely for universal algebra $\ $ [or even for first order structures].
Let's fix an algebraic signature consisting of named finitary operation symbols, and a variety $\mathcal{Alg}$ of algebras of that signature that we're working on. (In your question, the groups.)
As in your question, we use $F:\mathcal{Set}\to\mathcal{Alg}$ for the free functor and $U:\mathcal{Alg}\to\mathcal{Set}$ for the underlying set functor.
A presentation $\langle A,E\rangle$ is a pair where $A$ is a set and $E$ is a subset of equations, i.e. pairs of terms (in other words $E\subseteq F(A)^2$) $\ $ [or, $E$ is a set of atomic formulas].
In the case of groups, an equation can be arranged on one side, leaving the identity on the other side, so then $E\subseteq F(A)$ is enough, as learned.
We can define a morphism of presentations $\langle A,E\rangle\to\langle B,\Sigma\rangle$ to simply be a function $f:A\to B$ such that $$(\tau,\sigma)\in E\ \implies\ (Ff(\tau),\,Ff(\sigma))\in\Sigma\,.$$
By an interpretation of the presentation $\langle A,E\rangle$ in an algebraic structure $\mathfrak B$ we mean a function $h:A\to U\mathfrak B$ such that all elements of $E$ are evaluated equal $\,$ [all atomic formulas of $E$ are true], i.e. for every $(\tau,\sigma)\in E$ we have $$\varepsilon_{\mathfrak B}\left( Fh(\tau)\right) \ =\ \varepsilon_{\mathfrak B}\left( Fh(\sigma)\right)\ \text{ that is, }\ (\tau,\sigma)\in\ker(\varepsilon_{\mathfrak B}\circ Fh)$$ where $\varepsilon:FU\to 1_{\mathcal{Alg}}$ is the counit of the free-forgetful adjunction.
Observe that we can compose morphisms of presentations with interpretations as well as with homomorphisms, in an associative way, since all are just functions.
So this actually connects up the category of presentations $\mathcal{Pres}$ with the category of algebras $\mathcal{Alg}$ by (so called hetero-)morphisms between them, yielding to a bigger category, containing both $\mathcal{Pres}$ and $\mathcal{Alg}$ as a full subcategory.
Such an animal is called a profunctor, and it is determined by the restriction of its hom functor to the heteromorphisms, i.e. of a functor of the form $\mathcal{Pres}^{op}\times\mathcal{Alg}\to\mathcal{Set}$.
Now, every presentation $\langle A,E\rangle$ has a reflection in $\mathcal{Alg}$, namely $FA/(E)$ where $(E)$ is the congruence relation generated by $E$.
This defines the free functor on presentations.
Conversely, every algebraic structure $\mathfrak B$ has a coreflection in $\mathcal{Pres}$, namely $\langle U\mathfrak B,\Delta_{\mathfrak B}\rangle$ where $\Delta_{\mathfrak B}$ is the set of all equations that can be written on letters of $U\mathfrak B$ and hold in $\mathfrak B$ $\,$ [the positive diagram of $\mathfrak B$].
This defines the right adjoint.
Note that this is a fully faithful functor.
Giving a presentation $\langle S \mid R \rangle$ of a group amounts to describing it as the cokernel of a map $F(R) \to F(S)$ between free groups. There is a category $C$ whose objects are such maps and whose morphisms $(F(R_1) \to F(S_1)) \to (F(R_2) \to F(S_2))$ are maps $f : S_1 \to S_2$ such that the induced map $F(f) : F(S_1) \to F(S_2)$ sends every relation in $R_1$ to an element in the image of $F(R_2)$. Taking the cokernel to get the group is a functor on this category, which has a right adjoint sending a group $G$ to the map $1 \to F(G)$.
Explicitly this adjunction means the following: we have
$$\text{Hom}_{\text{Grp}}(\langle S \mid R \rangle, G) \cong \text{Hom}_C(F(R) \to F(S), 1 \to F(G))$$
which follows since both sides describe the set of functions $f : S \to G$ such that every element of $F(R)$ is sent to the identity.