Let $\Sigma$ be a set which consists of function symbols with allowed infinite arities (which can be arbitrary sets). A $\Sigma$-algebra is a set $X$ equipped with maps $\omega_* : X^d \to X$ for each function symbol $\omega \in \Sigma$ of arity $d$. We have a category of $\Sigma$-algebras with a forgetful functor to the category of sets.
Question. What is a reference for the construction of free $\Sigma$-algebras?
Nowadays it seems to be quite popular to define algebraic theories in such a way that free algebras are already included, more or less, into the definition. This is not the case with the definition above. Of course, the classical case is when the arities are finite sets. In this case, it is quite clear what to do. One defines terms of a given height inductively. A term of height $0$ is one of the chosen generators, and a term of height $\alpha+1$ is a function symbol paired with a family of terms of height $\leq \alpha$ (but at least one of height $\alpha$) indexed by the arity. If $\alpha$ is a limit ordinal, do we define a term of height $\alpha$ as a function symbol paired with a family of terms of unbounded heights $<\alpha$ indexed by the arity? Then, if $\alpha$ has cofinality larger than all arities, there is no such family and therefore there will be no term of height $\alpha$, and hence also no term of height $\geq \alpha$. Hence, the coproduct of all the sets of terms of a given height is well-defined, and I guess that this is the underlying set of the free $\Sigma$-algebra. Is this correct? References to the literature are highly appreciated.
You can attempt to define free algebras in the same way. It is the usual practice, as your first paragraph seems to suggest, to let the arities be cardinals, not ordinals, as you assume later on. Then if $A$ is a set of terms of cardinality $\beta$ and $f$ is a function symbol of arity $\beta$, we say the symbol $fA$ has rank the greater of $\beta$ and the supremum of ranks of terms $a\in A$. This disagrees mildly with your conventions.
The problem is that the class of terms so produced may not be a set. In fact, explicit examples have been known since 1964, most famously the free complete lattice on three generators. References include the wikipedia article on free lattices and its references, including Johnstone's Stone Spaces do this article, the link to which might not work.