What does 'as' mean when followed by v-ing?

The more interesting phrase here is "[In the context of Ns], N stands out as A", where N is a noun that is part of Ns, and A is an adjective or adjective-like phrase. Here are some examples:

• In the history of ethics Bentham stands out as one of the ablest champions of utilitarianism.
• The year 1880 stands out as one of manifold misfortune and disaster to agricultural Russia.
• You want to stand out as the best applicant the hiring committee has ever seen.
• Some of the great fairy tales stand out as answering to this test.

The phrase indicates that N is particularly A in the context. In your example:

1. What requires what? The "classes...require...approaches".
2. If you replace 'as requiring' with 'which require', it would imply that they really "stand out" more. A different phrasing closer to the original meaning might be "there seem to be three fundamental classes that clearly require different approaches".
3. as+v-ing is not a fruitful analysis in this case, both because "as" is part of the phrase "stands out as", and because "requiring different approaches" is adjective-like.