Is the set of all Context free languages a Context sensitive Language? ( can we build a LBA that decides whether a given language is CFL or not?)
If you consider a grammar $(N, \Sigma, P, S)$ you could actually give a regular expression for the production rules $P$, which pretty much determine the whole grammar:
$$r = S \to (\bar{N} + \bar{\Sigma})^\ast (| (\bar{N} + \bar{\Sigma})^\ast)^\ast (, \bar{N} \to (\bar{N} + \bar{\Sigma})^\ast (| (\bar{N} + \bar{\Sigma})^\ast)^\ast)^\ast$$
over the alphabet $N \cup \Sigma \cup \{,, |, \to\}$, where we use the $\bar{N}, \bar{\Sigma}$ as shortcut for $\sum_{A \in N} A$ and $\sum_{a \in \Sigma} a$.

So you could pass the grammar (the "code" of the cfl) to a finite automaton to check whether it is a context-free-grammar.

Basically, you just have to check whether there is only one single non-terminal in front of a $\to$. There is no nesting in contex-free grammars in contrast to regular expressions.