If Gideon uses 'Command' to let a Rebel Trooper attack, does this Trooper profit from his 'Aim' ability?

On the one hand it isn't his activation, but he wasn't moved either.

As long as the Rebel Trooper did not "exit the space during this activation", yes.

But when Gideon issues a command, it is Gideon's activation, and not the Rebel Trooper's activation. I don't have the card in front of me, but can Aim function at all when it is not actually the Rebel Trooper's activation?

It doesn't say anything about a specific activation just "this".

Are the Rebel Troopers firing durning Gideon's activation and is Gideon's activation "this"?

Yes to both so it works...unless Gideon also ordered that firing model to move first

Interesting. I wonder if that wording was intentional or an oversight; for example the rules on heroes taking strain for extra movement points specifically says "his activation" so they can't do it when getting an order from Gideon. But if the Trooper's Aim says "this activation" I guess that does imply that it's anyone's.