Is it possible for a Slayer to study a Target outside of battle to gain the bonuses for roleplaying purposes, rather than combat bonuses?
Yes, you can
Activating Studied Target is a move action, and there is nothing in the ability that restricts its use to combat situations. All actions are also possible out of combat (otherwise we couldn't even move around). As such, a slayer is allowed to walk around the city and declare that he is studying a guard, a merchant or any bystander, in order to benefit from the skill bonuses from Studied Target.