I am wondering how Greedo's "Slow on the Draw" ability works when he is controlled by the enemy. Say I use Lure of the Dark Side on Greedo, and target one of Greedo's allies. Do I get to make an attack on Greedo with that allied figure before Greedo shoots?

When a controlled figure is performing an attack its abilities are available, so I would say Slow on the Draw applies to the attack.

The figure that is being controlled is hostile to all figures and all figures are hostile towards it during the attack (and not friendly), so the target can also target Greedo for the Slow on the Draw attack.

You could use Lure of the Dark side to attack with Greedo, and that would trigger Greedo's slow on the draw. This would give the target of that attack a chance to attack Greedo first. This would be at the discretion of your opponent (since they controls the target of Greedo's declared attack) because Greedo's slow on the draw is based around the keyword "May". So the target can attack greedo, but is not obligated to do so. Your Lure of the darkside only applies to Greedo himself and not to Greedo's target, so you don't get to force the attack on Greedo. I would think in most cases your opponent would elect to not attack Greedo.

i.e. most cases are in 2-player skirmish, where your opponent would not choose to attack Greedo, he being the opponent's own figure. (Unless you control Greedo to attack your own figure - for example to get rid of his focused condition by attacking the Junk Droid - yeah, a very far-fetched scenario.) In 4-player free-for-all it depends on who Greedo is attacking.

OP did specify that Greedo was attacking his own ally, in which case opponent would probably choose not to shoot at their own Greedo.