Is there a specific relation between sideslip angle and yaw rate OR how sideslip angle depend on yaw rate?
Since, yaw rate is controlled via rudder


No, yaw is controller by many things, and rudder is there mainly to compensate for them to keep the plane flying nose-forward.

The usual way to make a control system is that you control (with a PID controller):


Move elevator to target vertical (plane coordinates) acceleration.
Move ailerons to target roll rate.
Move rudder to target zero lateral (plane coordinates) acceleration. Lateral acceleration is measured by the ball in traditional cockpit.


Note that each axis is basically independent and rudder is only used to eliminate side-slip, which is proportional to the lateral acceleration unless you have asymmetric thrust or drag—and I have not heard of a FBW that would automatically correct for those conditions.

When flying with auto-pilot, the auto-pilot sets vertical speed target from altitude error and vertical acceleration from vertical speed error for the pitch and it sets roll from heading error and roll rate from roll error for roll axis. Autopilots don't set lateral acceleation targets at all—that is always zero.

That is why auto-land has much lower cross-wind limit—autopilot does not handle rudder, so it can't de-crab.