Hello @Lokee,
The Battered Assaultron Head Laser is not a Walked Fire Attack but in your first point you did ask about Walked Fire. I likely just got the two questions mixed up. Let’s try again;
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Protectron and Walked Fire: They would roll for each point of damage.
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Battered Assaultron Head Laser: It would trigger on any action (other than a shooting action, note the ‘exc.shooting’ in the rules text), be it a standard Action or Quick Action (be in earned through Action Points or a Program: Spread).
You would trigger it on a Quick Action Move providing you have a non-friendly model in LOS. You would check for this once the action was completed, so you’d check after placing the model on the board as suggested in the rules text (it states ‘After each action’).
No stipulation is given to the range of LoS (see p17 of Rules of Play), just that a straight line must be measured from one part of the attacker’s base to the other without being blocked.