If a fighter jet veers off the runaway, is nosewheel steering needed to counter, or can directional control be maintained with the rudder deflection alone? Is there a chance that the rudder deflection alone won't be enough to change the aircraft direction? Is there a speed zone or gap were the rudder deflection is not aerodynamically available, while at the same time steering of the front landing gear is ineffective due to loss of friction? Or does the max speed for nosewheel steering and minimum speed for rudder effectiveness overlap each other?
There is some overlap in effectiveness, and therefore no real magic number, but most aircraft that don't have full time nosewheel steering will have some published airspeed at which one will be favored over the other. For example, in the EA-6B it was 80 KIAS. On take off roll you would hold the nosewheel steering button on the control stick down to maintain directional control with the nose wheel. Accellerating through 80 KIAS it was a call-out to release the NWS button and rely on the rudder effectiveness to keep control up to rotation at around 145 KIAS. The reason for this was not because the nosewheel was less effective due to less friction, (traction) but rather there was a risk of overcontrolling steering at higher speeds.
Yes, the steerable nosewheel and the rudder always work together. They both contribute a yaw torque in the same direction. As to your question about a speed regime where neither is effective-- I can't really answer that from first-hand knowledge across the entire aircraft design spectrum but it seems that the answer must be "no" in most cases or else the aircraft in question would not survive for long.