Evaluating a classifer that places instances into multiple (more than two) categories

Cardinal and nominal mean the same thing in this context. If the classes are well balanced, accuracy is fine, but if they aren't, you can measure precision/recall/f-measure for each class, and report a mean of those values overall (or a per-class value, if you have access to it).

For ordinal classes, it's the more complex problem of learning to rank, and the proper way to measure depends on the loss of the real world. Usually one evaluates a group of items at once, having a ML system produce a ranked list and computing a loss function on that list. If you really want to evaluate each example on its own, the techniques from the previous paragraph are appropriate. If not, there are many common ranking losses, such as precision@K, NDCG, and others that penalize the top of the list more strongly. If any element in any wrong position is just as bad, Kendall's tau is a common thing to report.