What are the gap functions in the $AC$ hierarchy? Hastad had in 1985 shown that PARITY(n) if it has to be evaluated by a depth$-d$ $AC^0$ circuit needs a size $\Theta(2^{n^{\frac{1}{d-1}}})$. But PARITY is in $NC^1$ and PARITY is also the negation of the MOD2 and hence this shows that the following inclusions are strict, $AC^0 \subset AC^0[2] \subset NC^1$. 


If I understand right then Razborov's 1987 result of exponential size lower bound for MAJORITY for $AC^0[2]$ or Smolensky's similar result in 1986 for MOD3 for $AC^0[2]$ do not in anyway strengthen the inclusion consequence of Hastad's result. Am I right?
But do we know whether $AC^i$ vs $AC^{i+1}$ separation is strict or not? Like do we know of functions which are exponentially sized for $AC^i$ but can be computed by polynomial sized circuits for $AC^{i+1}$?
Razborov-Smolensky prove that the classes $\mathrm{AC}^0[p]$ and $\mathrm{AC}^0[q]$ are incomparable when $p$ and $q$ are distinct primes, which also implies
$$\mathrm{AC}^0\subsetneq\mathrm{AC}^0[p]\subsetneq\mathrm{ACC}$$
for prime $p$. That's about it: it is consistent with the present knowledge that uniform $\mathrm{AC}^0[6]$ coincides with all of the polynomial hierarchy PH, so no classes in between have been unconditionally separated (uniformly or nonuniformly).