Chomsky hierarchy type determined by language I have some modified automata and the task is to give the type of Chomsky hierarchy to it.
All task is between type 3 and 0 noninclusive.
For regular languages there are lot of tools and I can check it without problems, Turing Machine equivalent is also easy task, and there will be no such examples.

Now the question: is it sufficient to show that automaton can accept specified language of given type?
From what I checked it would be sufficient to show equivalence to for example to NPDA, so I assume that if machine handles language that at least NPDA accepts it would be sufficient.

For example if machine can accept $a^nb^n$, it is type 2.
If machine can accept $a^nb^nc^nd^n$ it is type 1?
If not are there better examples of such languages or what steps should I follow?
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.