I've noticed air coming through each hole in the door frames where the deadbolt goes through,throughout the house. Is this supposed to be normal?

Nope.

You're probably going to need to pop the trim, maybe cut away a little drywall, and get in there to air seal.

Looks like whoever built or remodeled your home before you screwed up the detailing in the door framing.

You don't have to do any of that. Get a can of Great Stuff window and door straw foam at the Homarama. Drill a series of 1/4" holes in the jam. It's the flat part that your showing in your picture. Shot the foam in ,don't get crazy with it. It expands. Bend the straw up. The foam will flow from one hole to the next if you put them close enough to each other. Only takes an hour to dry. Cut it off with a razor blade or sheet rock knife. It takes paint, so brush a coat on and your done. It's quicker and better than chinking with fiberglass. I would check any other doors, if they did it to one they probably did it to others. You'll have extra foam anyways. Wear gloves, the stuff sticks to everything an can be hard to get up. Lay down newspaper. Acetone will clean it up but has to be used sparingly. It is a paint remover.

Thanks for the suggestions. We had a brand new a/c installed about 1 year ago and it isn't cooling as good as we thought it would. We had the a/c checked by several companies and they say it is fine. I have started checking for all of the possible air leaks in the house and have found several so far.

How old is the house and location? A blower door test might be a good idea.