If it has a Floyd Rose style bridge then that could be a big contributing factor. Floyd style bridges are only balanced by the mounting screws in the front of the bridge and the strings themselves. trying something like a tremolno to hold the bridge in place might get you more in the ball park...
Your action might be too low. With dropped tuning and with thicker strings they have a larger circumference when they vibrate. You could try raising the action a bit and see if it helps with the fret buzz. As far as the strings feeling too loose switching to a tune-o-matic bridge could help. It...