$\bar{c}=\sqrt{\frac{8}{3 \pi}} \cdot c_{ rms }$

$\Rightarrow \bar{c} < c_{ rms }$

So, statement $1$ is incorrect.

Mean free path of molecules is larger than mean distance between molecules.

So, statement $II$ is correct.

Mean free path decreases with temperature as collision frequency increases with temperature. So, statement $III$ is incorrect.

As, $c_{ rms } \propto \frac{1}{\sqrt{M}} \Rightarrow rms$ speed of nitrogen molecule is higher than that of oxygen molecule at same temperature. So, statement $IV$ is incorrect.