Is it possible for a Turing machine to be able to reduce a grammar and tell where it fits in chomsky hierarchy? For example:
This looks like a context free grammar:
 → 
 →  | 
 →  | 
 →  | c

but it can be reduced to this regular language:
 →  | 
 →  | 
 →  | c

I want to know if it is possible or not and why
It is not possible: it is undecidable whether a context-free grammar describes a regular language. For a proof, see e.g. Undecidable Problems for Context-free Grammars by Hendrik-Jan Hoogeboom.