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The compendium from Crescenzi is the most complete that I am aware of, despite being pretty old now. You can also watch this more recent (still twelve years old) survey: Vangelis Paschos. An overview on polynomial approximation of NP-hard problems. https://hal.archives-ouvertes.fr/hal-00186549/document


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Since it is asymptotic approximation and epsilon is a constant, for OPT big enough being 1 off is always good. Let's put it another way. Either your optimal is smaller than 1/epsilon and you can find it within polynomial time. Or it is not and thus 1+1/OPT is better than 1+epsilon.


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The following website seems to be no longer maintained, but it is still a useful resource because it covers many problems: http://www.csc.kth.se/~viggo/problemlist/


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Not a complete answer, but hopefully a good starting point. It is very instructive to (always!) first consider the discrete analog of your question. If $X$ is some set and $f:X\to\{0,1\}$, what is the minimal number of evaluation queries needed to uniquely identify $f$? As already noted in the OP, the question only makes sense if one fixes a function class $...


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