If I cut the same material multiple times to I have to z probe the board before every cut path or will the computer remember?
The computer will remember your home position and your Z height, just select manual when it ask you how are you setting your Z.
If you homed your machine before setting your work zero the work zero is persistent between power cycles also. If you change the bit only Z need to be re-zeroed
If work zero is set without homing the machine the work zero is persistent as long as the machine have power/stay connected via USB (signalwise). If you change the bit only Z need to be re-zeroed