Chomsky hierarchy has four types of languages and grammars.
If we have some language $L$, what are the tools for finding out the correct family of languages it belongs?
I know that there are pumping lemma's that we can show for example that L is not regular language and therefore it doesn't belong in regular language family.
What other tools are there?
Are there tools for generating a grammar for certain language?
Example language:
$$L:\{0^{n}10^{n+4}\mid n\geq 1 \}. $$
We want to know in which family $L$ belongs.
How we do it in correct way?
Your $L$ differs only slightly from the classic example $\{0^n1^n\mid n\ge 1\}$ of a context free language (CFL) that is not regular. The proof of non-regularity is by the pumping lemma, and is easily adapted to this $L$. It's also easy to show that $L$ is a CFL.
Worth mentioning is the Myhill-Nerode Theorem, which characterizes the regular languages, and also provides a powerful tool for showing that a language is or isn't regular.
There is also a pumping lemma for CFLs, which can be used to prove that a language is not context-free.
Each level of the Chomsky hierarchy can be characterized in multiple ways — by grammars of a certain form, as well as by automata of a particular type. A language can be shown to belong to a certain level of the hierarchy by exhibiting an appropriate grammar that generates it, or an appropriate automaton that recognizes it.
Between the top two levels of the Chomsky hierarchy is a class of languages which really deserve attention: the decidable (aka recursive) languages. They're a proper subset of the "type 3" languages, which are generated by general rewrite systems. The type 3 languages are precisely the semi-decidable (aka recursively enumerable, or r.e.) languages. The decidable languages are type 3 languages whose complements are also type 3 languages. These aren't distinguished from the semi-decidable languages by the form of the grammars that generate them, so they don't have a "level" of their own. In computability theory (aka recursion theory) there are various techniques for showing that languages are decidable, semi-decidable but not decidable, not even semi-decidable, and so on to even further degrees of (non)computability.