Risa

And yeah why do we never hear about the shattered tanks in hydroponics, people falling into the warp core, and doctors getting stabbed by their own scalpels every time a laser hits the ship?

Generally this is glossed over. First off, red alert procedures mean preparing for battle conditions, so it probably means halting non-essential operations that would be high risk during a battle, as well as increased precautions by essential personnel.

Second, we get immediate damage and casualty reports, which includes a fly by mention of damage and injured crew. That probably covers most of what you are thinking of. It's all repaired by the next episode anyway.