In which situations is it possible to construct a coproduct this way?

Let $\left(F,U;\eta,\varepsilon\right):\mathcal{\mathcal{X}\rightharpoonup A}$ denote an adjunction where $U$ is faithful and where $\mathcal{X}$ is a category that has coproducts.

You can think of $U$ as underlying functor and $F$ as free functor.

I made an effort to construct a coproduct for a family of objects $\left(a_{j}\right)_{j\in J}$ in $\mathcal{A}$.

I started with a coproduct $x:=\sqcup_{j}Ua_{j}$ in $\mathcal{X}$ equipped with injections $\iota_{j}:Ua_{j}\to x$.

Since functor $F$ is left adjoint it will preserve this coproduct so $Fx\in\mathcal{A}$ equipped with injections $F\iota_{j}$ will serve as coproduct for the objects $FUa_{j}\in\mathcal{A}$.

We have the components $\varepsilon_{a_{j}}:FUa_{j}\to a_{j}$ of the co-unit and the fact that $U$ is faithful ensures that these arrows in $\mathcal{A}$ are epimorphisms.

In quite some situations (e.g. if $\mathcal{A}$ denotes the category of groups or monoids and $U$ is the forgetful functor to the category of sets) it is possible to construct some quotient $Fx/\sim\in\mathcal{A}$ together with an arrow $\nu:Fx\to Fx/\sim$ and arrows $\delta_{j}:a_{j}\to Fx/\sim$ such that the following diagrams commute for each $i\in J$ and such that $Fx/\sim$ equipped with injections $\delta_{j}$ indeed serves as a coproduct of the $\left(a_{j}\right)_{j\in J}$.

$$\begin{array}{ccc} FUa_{i} & \stackrel{F\iota_{i}}{\longrightarrow} & Fx=F\left(\sqcup_{j}Ua_{j}\right)\\ \varepsilon_{a_{i}}\downarrow & & \downarrow\nu\\ a_{i} & \stackrel{\delta_{i}}{\longrightarrow} & Fx/\sim\end{array}$$

I have not the impression that this construction will always work. For instance I really needed arrows that could be recognized as maps.

So automatically the question rose: when will it work?

My questions are:

Are there specific characterics for the situation in which the construction works?

If you recognize this construction then can you provide references?

As you say, you can't do anything like this with an arbitrary adjunction, but the examples you mention are very special cases: they are monadic over $\mathrm{Set}$, and in that case such a construction always works. Basically you use the fact that every object $a ∈ \mathcal A$ is the coequalizer of maps $εFU, FUε : FUFUa → FUa$, so you'll want to take $ν$ to be the coequalizer of $\coprod_i FUε, \coprod_i εFU : \coprod_i FUFUa_i → \coprod_iFUa_i$, and $δ_i$'s will be the induced maps between the coequalizer diagrams.
To do this, you need $\mathcal A$ to have coequalizers (in fact, reflexive coequalziers are enough), which it does if $\mathcal X$ if $\mathrm{Set}$, but not in general. Fortunately there is a number of conditions to ensure that the coequalizers you need will exist; see nLab, or the second volume of Borceux's Handbook of Categorical Algebra for details. If you want the map $ν$ to really be a quotient by an (appropriate generalization of an) equivalence relation, and not just a coequalizer, you'll probably want $\mathcal A$ to be regular or exact; a sufficient condition for that is listed in both references.