Can you use Duke to investigate an unrevealed location?

Unrevealed locations have no clues and no Shroud value.  An unrevealed location only becomes revealed when instructed or the first time an investigator moved into that location.  He does not allow you to investigate an unrevealed location.  Though you can move to a location with Duke.  It would then become revealed.  Then you can investigate it.  The move happens first.  therefore the location would be revealed.  So effectively yes the location can be unrevealed before you use Duke's action.

Do flat stat increases like Beat Cop's +1 Fight add to Duke's fight check? Or is the only way to boost those specific actions with Skill cards and paying for Talents after the checks have started?

Duke changes your base stat.  This basically means Duke's stat replaces Pete's stat.  Then passive bonuses like Beat Cop, Dr. Chris... can improve the check. 

 

Though since Duke's ability requires an action to use you can't combo him with thing like .45 or Flashlight since those abilities would also require an action to initiate.

When you use Duke's Investigate ability, you are allowed to first move to a connected location. As stated in another post here, that location can be unrevealed, since moving there will reveal it. Now, what happens when you use Investigate on Duke, move to an unrevealed location and a Forced effect there has an Enemy spawn at that location that engages you. Are you obliged/committed to execute the Investigate action that you technically are in the middle of? This would mean an unforeseen Attack of Opportunity that is unavoidable, right? Or does the Attack of Opportunity not trigger? The Enemy did not exist when you made the choice to Investigate, so penalizing that decision seems off in a way?

You paid the cost for the action, so now you must finish it if possible.

However, Attacks of Opportunity only trigger when you are paying the costs for an action that would trigger them (technically they happen immediately after the costs, but before the effect). You've already paid the cost for Duke's action by exhausting him and spending the action, so any enemy that somehow became engaged with you as you move into that location will not have an AOO.

Does Duke counts towards ally count for ally slots? I mean, If I have the stray cat in my hand and Duke is on table, Can I deploy the stray cat still? 

 

Thanks!!

Duke has the Ally trait, but he doesn't have the "Ally" icon in the bottom corner of the card.  That is where the restriction comes from.  Duke does not count.  So yes, Duke is cat friendly.

He does not.

If you look at the Stray Cat, there is a little picture of a shadowy guy in a circle down the bottom right, that is the Ally Slot icon/symbol. Only things with a slot symbol on them take up your slot (you will find most scenario specific allies don't have it, while most earn-able campaign allies do - compare Adam Lynch or Innocent Revelers to Lita Chantler).

 

Armed with this knowledge we find that while Duke has the Ally trait, he does not have the Ally slot symbol and thus doesn't stop you playing other Allies.