Assuming $\mbox{BPP}\subseteq \Pi_2$ -What conclusions can we make?
It is known.  As your final statement says, $\mbox{BPP} \subseteq \Pi_2 \cap \Sigma_2$.  It is called the Sipser–Gács–Lautemann theorem.  (All though your "so" is somewhat misleading.)

It can be strengthened to $\mbox{BPP} \subseteq \mbox{MA} \subseteq \mbox{S}^P_2 \subseteq \Pi_2 \cap \Sigma_2 \subseteq \Pi_2$ (see also Arthur-Merlin protocol).