Have you looked at the torch on/off indicator in Mach 3 (see attached image). I you run your routine and watch that indicator if the torch is on it will blink and stay out when traveling to the next cut. If I understand what you said it turns on the torch for the first cut and off after it reaches the location of the second cut on point. And stays off for the second torch cut and stays off until it gets to the third cut then turns on. You can run the routine with out running the table and it will go through the routine you can watch the torch indicator to see if it cutting as you want or it may be doing what the g-code told it to do. This test will point to where the problem is. If the indicator blinks as you described the g-code is telling it to do that. If it blinks only when you wanted it to cut then you may have a switch problem.
Are you able to take a screen shot of the g-code and attach it here like this.
Everlast PM256
Millermatic 180
Hypertherm PowerMax 65 with machine torch
Longevity Force Cut 80I
DIY CNC table for plasma/routing
13" metal lathe
Small Mill
ect, ect.