A lot of you are suggesting physical excercise. While I agree, it's to an extent. Remember, this is a great dane the OP is talking about. Too much physical excercise can ruin their joints. A walk is good, even a small hike. But pulling a cart? Not good for this breed
Another option would be to teach the dog to scent out treats and eventually search for other things, scent work is very mentally stimulating to dogs. You could also teach them as many tricks as possible.
My dog who has anxiety and ptsd from a house fire very much enjoys learning new things and scent work and it has helped her to come out of her shell and moved past thing.
To the OP:
And when it comes to teaching the dog to be ok when you or someone leaves start with teaching them to stay in one spot, maybe get one of those dog beds thats raised up off the ground and use that as his "place" and just teach the command stay or wait or place whatever you want the word to be.
Then once they have that down put them in there "place" and start conditioning them to someone getting up from the couch and rewarding them for staying. Start with baby steps. Then reward them for staying when the person gets up and takes a step. (Mind you after each reward the person goes back to sitting) and keep going till the person can walk out of the room.
Then you keep going one step at a time, I'd say each step (like getting up, taking one step taking two steps etc..) should each have its own day. If the dog reacts at any point go back a step. If you cant even get off the couch without the dog reacting then start with just the motion like you were going to get up.
Might be best to try it with 2 people, one person to do the rewarding and the other person to do the leaving the room part.
Eventually when the dog gets better with someone leaving the room, then have the person stay out of the room for longer and longer.
Then when the dog gets better with that, then you can move on to leaving the house. When you do that though I would start over with getting off the couch and do step by step with that motion as well.
Sorry for the long post, hard to convey what I mean in a short post.