Which test to use to calculate correlations between them?
- Cramer's V,
- or something else?
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Cramér's V and the Kruskal–Wallis test are for nominal data; the latter is a null hypothesis test, not a correlation. If you want to calculate the correlation between a dichotomous variable and an ordinal variable, you could use Kendall's $\tau$, the Goodman–Kruskal $\gamma$, or Spearman's $\rho$ (listed in the order in which I'd recommend them, I suppose). If you're estimating population parameters, you can also generate confidence intervals around these statistics or perform null hypothesis tests if you wish.
For some comparisons and a conversion method, see: