Metric induced by a norm - what conditions should this metric meet?

Metric induced by a norm - what conditions should this metric meet?

$(I)$
I've been browsing some problems concerning metrics not induced by norms,
and I've found a comment that said that such a metric should be a concave
monotone function. Here is the post I'm reffering to.
Could you tell me why that is?
$(II)$
I know that if a metric $d: X \times X \rightarrow \mathbb{R}$ ($X$ is a
vector space with scalars in $\mathbb{K}$ ) satisfies:
$(1) d(\lambda x, \lambda y) = |\lambda| d(x,y) \ \ \ \forall x,y \in X, \
\lambda \in \mathbb{K}$, in particular
$d(\lambda x,0) = |\lambda| d(x,0) = |\lambda| \ ||x||$
$(2) d(x+w, y+w) = d(x,y) \ \ \forall x,y,w \in X$, in particular
$d(x-y,0)=d(x,y)$
then the function $||u||:= d(u,0)$ is a norm. I have problems proving the
converse, meaning that if a metric defines a norm, then it satisfies the
two conditions $(1), \ (2)$.
Could you help me with that, too?
Is it true to say that if we want a metric to be induced by a norm it
should satisfy both conditions mentioned above?
I would really appreciate all your insight.