Nice work, you have made great progress.
The unwanted lines are clearly the laser not turning OFF correctly but the missing lines are strange. Perhaps you could observe the engraving to determine if the laser is turning OFF too early or turning ON too late ?
It could be related to your CAD/CAM but if you post your problem Gcode I will test it here and that will prove one way or the other.
The Mach3 Impact / Laser plugin only operates using the Parallel Port and will not work when using the ESS.
Tweakie.