What's up with the gravity on board of Bebop?
The Bebop uses centrifugal force to generate "gravity". It has a rotating cylinder which is where the crew spend much of their time when in space. It is shown from both inside the ship and outside the ship, and in a few episodes, it is shown that when the cylinder stops spinning, the people inside that portion of the ship suddenly start to float.

Presumably, there would be no apparent gravity in the portions of the ship which do not rotate when the ship is not thrusting. When the ship is thrusting (accelerating), then the entire ship would experience a force similar to gravity along the axis of thrust.