When Grogu flees from the planet, he jumps from his cradle right into the cockpit, closes the wind shield and takes off. Take off shot should show the cradle left behind, but there is none. Neither Grogu nor R5 have time to store it and there is no place for it inside or outside the ship anyway. Later when Grogu is inside Bo-Katan's ship, he is again in his cradle.
When Din releases the astromech to the surface of Mandalore, it is implied that the astromech shaft has an bottom opening right behind the central engine. But the astromech shaft is right above the front half of the engine so can't have a bottom opening as it would go right through the engine. Even the original Naboo fighter with no central engine has no bottom opening for the astromech shaft, so astromechs must enter and exit the shaft by crane or in case of emergency by their own rockets.
The Mandalorian helmet has no connection to the suit. It is put on and off freely. So it is open to the atmosphere and can't be pressurized. And even if there were a connection, the suit has no air supply.
Din Djarin tells Grogu to seal himself in his pod while he goes to find the astromech. He opens the cockpit, immediately exposing it to the atmosphere of the planet. Moments later Grogu appears again, proving that the air on Mandalore is breathable.
Bo-Katan praises Grogu twice for navigating the ship while ignoring the astromech whose main function is to navigate and autopilot ships.