Chomsky hierarchy type determined by language
No, a finite set of languages does not (usually) characterise infinite classes of languages. So any such specific tests are probably easy to fool.

For CFL in particular, there the Chomsky-Schützenberger characterisation. So if your model can deal with finitely many types of nested parentheses, it's probably at least as powerful as NPDA.

The generic proof technique for showing that one model is more powerful than another is simulation. You can find some examples via simulation and, of course, in the literature.

Proof of the opposite is usually done by counter-example, i.e. a language that can be expressed in one but not the other model.