Statistical test for finding relationship between an independent variable measured on an ordinal scale and a dependent variable measured on a nominal scale.
There is no such thing as a dependent nominal variable since nominal simply is used to assign categories. But ordinal variables are usually analysed using non-parametric tests (e.g. Chi Square) since they are based on responder inputs and are not normal in distribution :)
You need to specify the options of the nominal scale. For instance, if there is a two option or categorical data as against ordinal scale, you can use point bi-serial if they are normally distribute. if not normally distribute, you can use chi square.
The information in the question are insufficient. But if the dependent (nominal) variable could take binary outcome, such as the presence or absence of a feature, logistic regression would be appropriate. Chi-square could serve your purpose, but would depend on your distribution in the cell counts
Yes, when the response takes on dichotomous level (yes/no), logistic regression is preferred. If more than two categories (of no natural order), multinomial logistic regression is the choice, but if natural order exist, ordinal logistic regression will do well (provided the assumptions of proportional odds are met). Chi-square could be used, but it depends on what you are interested in. Chi square could tell you the 'power' of association between the two variables (nominal dependent and ordinal independent), but may not tell you the predictive 'power' with confidence intervals, which these other regressions could. In addition, whether you take the ordinal independent as dummy (or factor) variable also depend if the order is not large (say less than or equal to 5-7), when greater than say 7, it is good to take it as a scale (continuous) variable.
Logistic regression is more appropriate. In case your DV is binary, then you need to opt for binomial logistic regression; otherwise multinomial is the best fit.