AS to your first question concerning "reasonably believes", I'd say that would be subjective, and depends on the officer you encounter and why you are encountering him/her. If the officer is responding to a MWG call they may want to disarm you until they can assertain that you have a valid permit and aren't up to no good. Or they might make a big scene out of it and draw their weapon on you andn do the whole takedown BS to disarm you and THEN ask you to produce a HCP. Like I said it depends on the officer.
If you are pulled over for a traffic stop, it's best to let the officer know you are armed and produce your HCP right from the get go. (LEO's don't like surpirses) Since they probably don't know you from a fence post they probably will dissarm you until they are finished with the stop (unless they've caught you DUI). But this still depends on the officer that pulls you over. Some are curtious and proffesional while others may be asses.
Your second question, We have to have a HCP to carry, and in my opinion, LE wouldn't be doing their job if they didn't check to see if you have a permit to carry legally. How they go about this will depend on the officer.
Your third question I can't answer. I'm still waiting for my HCP so haven't started carrying yet. From what I've found out, in asking some of the local businesses here I shouldn't have any problems, but we'll see.