Classes containing boolean closure of CFLs
This is not an answer to your question per se, but might explain why finding sensible classes of languages more or less fitting your requirements will be hard: recall that the emptiness of intersection of two context-free languages is undecidable, thus the full boolean closure of CFLs is not a very interesting class... (In fact, any R.E. language $L$ can be expressed as $h(L_1\cap L_2)$ for $L_1$ and $L_2$ deterministic context-free and $h$ an homomorphism.)

More in the scope of your question, you can look at Alexander Okhotin's work on conjunctive and boolean grammars, but I don't recall any pumping results.