When asked by Team A head coach for a stick check on opponent during dead ball, sidewall of head was found to be broken and taped. No penalty given, but removed from the game. Was this correct? I reffered to rule 1-8 of NFHS, a broken crosse is considered no crosse.

Secondly, during the above stick check, Team B head coach asked for a stick check on Team A. Do you administer second request immediately or at next dead ball?

I believe you ruled correctly on the first stick check. As for the second request I would do it immediately.