For completion's sake, let me add a slightly more general concept: scaled spaces.

Let $M$ be a set, let $X$ be a totally ordered set and let $0$ be a symbol such that $0<x$ for all $x\in X$. An $X$-valued scale on $M$ is a map $d:M\times M\to X\cup\{0\}$ such that for all $x,y,z\in X$:

1. $d(x,y)=0\Leftrightarrow x=y$;
2. $d(x,y)=d(y,x)$; and
3. $d(x,z)\leq\max\{d(x,y),d(y,z)\}$.

The space $(M,X,d)$ is called a scaled space (ultrametric space if $X\subset(0,\infty)$).

This concept was studied by H. Ochsenius and W. H. Schikhof and then applied to the study of Banach spaces over fields with an infinite rank valuation. To initiate in this topic I recommend three articles:

1. H. Ochsenius and W. H. Schikhof, “Banach spaces over fields with an infinite rank valuation,” p-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 207, 233–293 (Marcel Dekker, 1999).
2. H. Ochsenius and W. H. Schikhof, “Norm Hilbert spaces over Krull valued fields,” Indagat. Math. 17 (1), 65–84 (2006).
3. A. Barria Comicheo, Generalized Open Mapping Theorem for $X$-normed spaces, p-Adic Numbers, Ultrametric Analysis and Applications, vol. 11, (2), 2019, pp. 135--150.

For completion's sake and regarding Question 1, let me add a slightly more general concept: scaled spaces.

Let $M$ be a set, let $X$ be a totally ordered set and let $0$ be a symbol such that $0<x$ for all $x\in X$. An $X$-valued scale on $M$ is a map $d:M\times M\to X\cup\{0\}$ such that for all $x,y,z\in X$:

1. $d(x,y)=0\Leftrightarrow x=y$;
2. $d(x,y)=d(y,x)$; and
3. $d(x,z)\leq\max\{d(x,y),d(y,z)\}$.

The space $(M,X,d)$ is called a scaled space (ultrametric space if $X\subset(0,\infty)$).

This concept was studied by H. Ochsenius and W. H. Schikhof and then applied to the study of Banach spaces over fields with an infinite rank valuation. To initiate in this topic I recommend three articles:

1. H. Ochsenius and W. H. Schikhof, “Banach spaces over fields with an infinite rank valuation,” p-Adic Functional Analysis, Lecture Notes in Pure and Appl. Math. 207, 233–293 (Marcel Dekker, 1999).
2. H. Ochsenius and W. H. Schikhof, “Norm Hilbert spaces over Krull valued fields,” Indagat. Math. 17 (1), 65–84 (2006).
3. A. Barria Comicheo, Generalized Open Mapping Theorem for $X$-normed spaces, p-Adic Numbers, Ultrametric Analysis and Applications, vol. 11, (2), 2019, pp. 135--150.

Regarding Question 2, in the context of scaled spaces, it was proved in [1] that a scaled space $(M,X,d)$ is ultrametrizable if and only if $M$ is discrete or there exist $s_1>s_2>\dots$ in $X$ such that $\lim_n s_n=0$.