Can Murne use Labored Attack while using False Orders on a Gamorrean Guard? If she does, does the Guard suffer strain, or does she?

Yes, she can use Labored Attack while using False Orders on a Gamorrean Guard.

The Guard suffers the strain.

Labored Attack is an ability of the figure being controlled, so it can be used during the attack. The "you" refers to the figure having the ability, so the Gamorrean Guard suffers the strain.

I don't understand the point with Labored Attack and strain suffering in general with non-heroes figures. They can't get Strain so they will take damage. Why not take damage in the first place ?

"Labored Attack : While attacking, you may suffer 1 damage to reroll 1 attack die" ?

Because when suffering strain, in skirmish you can choose to discard a command card from the top of the command card deck instead.

When suffering damage there is no such option.

Ok, skirmish. That makes sense. I only play Campaign so I forgot that rule