See it all of course not.
But in three weeks you can definitely see all the important stuff and some of the less known attractions. Also 5 grand for three weeks? Norway is expensive but I'm pretty sure you can do it for half that!
Unless going to TOS first is necessary for some reason, I'd fly into the cheapest destination between LON, CDG, or FCO and go from there. Plan 4/5 days per city and you'll have time to enjoy each destination.