I was born and raised in the south of Sweden and have been to Copenhagen a few times.
Copenhagen is just as safe as any city. Beware of pickpockets and if you go to Chrisitania, you need to chill and not offend anyone. I doubt they will have a problem with your camera if you seem like a nice guy. So don´t go around taking pictures - ask for them.
I second the Round tower and the canals. The mermaid (lille havefru) may be a must see but I´m not that impressed.
I´d take a bus or train to a small village by the coast or something to see the countryside instead of going to Malmö or Lund. Although they are nice citys aswell, they are pretty much the same anyway. Unless you are there when Malmöfestivalen is on, that is pretty nice and you should go. http://malmofestivalen.se/about/in-english
(16-23 of august). Lots of music and food from all over the world.
Sweden is also quite expensive when it comes to camera gear but without VAT it gets better. Also I think Sigma lenses are cheaper than ordinary, if that´s what you are into.
Personaly I use small primes when shooting in the street. 100 /2 could be perfect, as would 24 /2.8 IS and you should absolutely bring your 50 /1.8! Although the 70-200 /4 is the smallest of the whites, it´s still white. Drop the battery grip aswell. You want your camera small. Only models like to have big cameras in their face.
Remember to have smörrebrö and pölse. And of course snaps and beer. The Danish are great designers and almost as good looking as the Swedes, who are almost as good looking as the Norwegians.
Have a great trip.