Metrizable Topological Space In Many Ways

The statement is false: a set with a single element admits one and only one metric.

If we assume that $X$ has more than one element, then, although your proof works, I think that it is simpler to say that, if $d$ is a metric on $X$, then, for each $k>0$, $kd$ is another metric on $X$ which induces the same topology.

You proved that open sets in $d$ are open in $d'$. But you also have to prove the converse. For this you just have to replace $\frac {\epsilon} {1+\epsilon}$ by $\frac {\epsilon} {1-\epsilon}$ in your argument (taking $\epsilon <1)$. Except for this your construction of the metrics is is fine.