You are describing free constructions between finitary varieties.

A finitary variety is an equationally defined class of algebras for (i) an arbitrary set $\Sigma$ of operation symbols each $\sigma \in \Sigma$ having finite arity, (ii) an arbitrary set of equations $E$ consisting of pairs $(\phi_1,\phi_2)$ where each $\phi_i$ is a term built from the operation symbols and some fixed countable set of variables.

Then the induced finitary variety $\mathcal{V}$ is a category. Its objects are sets equipped with the operations from $\Sigma$ that satisfy the equations $E$. Its morphisms are those functions between the carriers that preserve each operation in $\Sigma$. Composition is the usual composition of functions.

Now, suppose $(\Sigma_1,E_1)$ specify the variety $\mathcal{V}_1$ and $(\Sigma_2,E_2)$ specify the variety $\mathcal{V}_2$.

In the case where $\Sigma_1 \subseteq \Sigma_2$ and $E_1 \subseteq E_2$, then there is a free construction $F : \mathcal{V}_1 \to \mathcal{V}_2$ which is the left adjoint of (i.e. uniquely determined by) the forgetful functor $U : \mathcal{V}_2 \to \mathcal{V}_1$ which merely forgets the additional operations.

Examples:

0. Let $(\Sigma_1,E_1)$ be the usual axiomatisation of monoids, so that $\mathcal{V}_1$ is the variety of monoids. Let $\Sigma_2$ also contain the additional semiring operations (you are adding +,0) and $E_2$ also contain the relevant equations (e.g. that + is a commutative monoid). Then the induced free functor $F : \mathcal{V}_1 \to \mathcal{V}_2$ is what you describe: it constructs a free commutative monoid and forces the original monoid to distribute over it in the 'simplest' way.

1. One can do the analogous thing but now with abelian groups and commutative rings. As the above comment states, the latter are not the same thing as fields (which do not form a variety in the sense above).

2. Similarly one can go from monoids to idempotent semirings.

3. Your third example again follows: you are extending the signature and operations.

Two more things.

Firstly the free algebra construction $F : \mathsf{Set} \to \mathcal{V}$ is a special case: the category of sets is the finitary variety with no operations and no equations. So for example, the free monoid construction is covered.

Secondly the conditions $\Sigma_1 \subseteq \Sigma_2$ and $E_1 \subseteq E_2$ are a special case of a more general construction: a translation between algebraic theories, which again induces a unique free construction $F$ as above.