After stage separation, why does the booster eject so many short thrusts?
One of the many, many ways that launches can result in failure occurs shortly after stage separation when residual fuel in the separated stage causes that lower stage to collide with the upper stage. SpaceX itself suffered this failure in 2008. To avoid this failure, the lower stage needs to quickly point itself away from the upper stage. This direction change makes the lower stage fly past the upper stage rather than collide with it if residual fuel remains and happens to ignite.
But why are they so "pulsative" and not in a more continuous manner?
Attitude control thrusters typically are either on or off, with no in-between. (This is quite typical. The heater and air conditioner in your house or appartment most likely function on the same basis: On or off, with no in-between.) This on or off behavior is directed by a bang-bang controller.
Concerning the same short thrusts: Can anyone explain the sequence of their directions?
The thrusters are fired in whatever the attitude controller deems necessary. Bang-bang control is fairly simple: There's a direction in which the vehicle is supposed to point (plus or minus a deadband), or a change in direction in which the vehicle is supposed move (plus or minus a deadband). Nothing happens when the attitude / attitude rates remain within these headbands. When the attitude error or attitude rate error goes outside the deadbands, the controller activates thrusters based on a jet select algorithm.