Choosing appropriate statistical test for ordinal data
You can use ordinal multinomial regression (also known as ordered logit) if the response is ordered. These methods are basically extensions of logistic regressions, but using e.g. a cumulative logit instead of the logit. However, there are a number of different assumptions you need to consider. For instance, are you going to use a proportional-odds assumption (which is commonly used), which means that there is an equal probability of going from e.g. class 1 -> 2 and class 5 -> 6? You can also evaluate the proportional odds assumption using plots or a score test. If the response cannot be ordered, there are multinomial nominal methods that you can use, and you can also evaluate the results from an ordered analysis by comparing the predictions with ones from a multinomial nominal analysis. I have used these kinds of methods for the analysis is Red list classifications, which are clearly ordered but cannot simply be transformed into a numerical response (similar to your situation).

The book Analysis of Ordinal Categorical Data (Agresti. 2010) is a really good starting point.

In R you can look at the packages polr and vgam for ways to perform different types of analyses on ordinal data. The author of the book above has also published some examples for categorical data analysis in R, using the packages I mentioned: Examples of Using R for Modeling Ordinal Data. In SAS similar analyses can be done using Proc Genmod and Proc Logistic.