The exact line from the rulebook is:
"They cannot fire if the vehicle moves Flat Out or uses Smoke Launchers that turn, nor can a vehicle move Flat Out or use Smoke Launchers if a unit embarked inside it shoots out."
SO its fairly straight forward that shooting and Flat Out in the same turn is a no-no.
On the second questions, I would say yes also.
Failing a Dangerous Terrain test causes one Hull Point PLUS an Immbolised result. If already Immobilised, it takes an extra Hull Point instead.
And the Drop Pod Immbolie rule does say (or at least the Blood Angels version): "and counts in all respects as a vehicle that has suffered an Immobilised result...."
And the inertial Guidance only saves it from Impassible Terrain, not Dangerous.
So, yes.