Let $(L,<,+)$ be a structure such that (1) $<$ is a linear order of $L$, (2) $L$ has a least element 0, (3) $+$ is a binary function on $L$ that behaves like addition of positive real numbers, i.e. commutativity, associativity, $a+0 = a$, and $a< b$ iff there is $c>0$ such that $b =a+c$.


Say a pair $(X,d)$ is an “$L$-metric space” if $d : X^2 \to L$ is a function satisfying the metric space axioms (only the codomain is not necessarily $\mathbb R$).


Question 1: Has this or something similar been studied?


Question 2: Say a topological space is “weakly metrizable” when there exists $L$ and $d$ as above inducing the topology. When is a space weakly metrizable?



A generalization of metric spaces
It follows from your assumptions that for $a<b\in L$ there is a unique $c$ such that $a+c=b$ and that $L$ is a cancellative monoid: $a+c=b+c$ implies $a=b$. Also addition preserves the order. A cancellative commutative monoid embeds in an abelian group. If I'm not mistaken $L$ embeds in an ordered abelian group $\Lambda$ such that
the positive elements correspond to the non-zero elements of $L$.


Metric spaces where the codomain of the distance function is an ordered abelian group - that is, $\Lambda$-metric spaces - have been defined as a first step to defining $\Lambda$-trees. Ian Chiswell has written an introduction to this subject.


As for the topology that such a metric gives rise to, note that if $\Lambda$ has a minimum non-zero convex subgroup, then this subgroup is archimedean, and thus the topology is $\mathbb{R}$-metrisable. If there is no such convex subgroup, then the balls of radius $\Lambda\_0$ - that is, the sets $B(x,\Lambda\_0)=\{y\in X:d(x,y)\in\Lambda\_0\}$ - where $\Lambda\_0$ ranges through the non-zero convex subgroups, gives a base for the topology which is therefore $0$-dimensional. And in fact replacing $\Lambda$ by the (linearly ordered) set of its convex subgroups and putting $d'(x,y)$ equal to the convex subgroup spanned by $d(x,y)$ gives a generalised ultrametric equivalent to $d$.


Generalised ultrametric spaces have been studied for example by Priess-Crampe and Ribenboim. (Of course if $L\subseteq\mathbb{R}\_{\geq 0}$, then these are ultrametric spaces.)


