Long story short. If my opponent uses the Lion Stronghold's ability to buff a character and then I steal it with Blackmail, does it keep the buff while on my side?

Yes, the +1 is applied to the characters they control at the time of the ability and persists until the end of the conflict. If you would take control of that character, it would still have the buff. Think of actions like this as a point in time bonus granted, and you don't need to revisit the play restrictions to see if they still qualify. Once the action is done, the bonus is granted. Likewise, if the opponent moved a character into the conflict after using the stronghold's ability, those characters would not get the buff as they missed it when the action was played.