I was born and raised in the Copenhagen area and has lived there for about 35 years, so I'm somewhat qualified to answer.
In general, Copenhagen and Denmark is quite safe for bringing gear. However, central Copenhagen is plagued by eastern European pickpocket gangs and professional beggars. Virtually everyone speaks English, so getting around and getting help should be easy.
You may feel ripped off every time you purchase something, but that's a separate topic best debated somewhere else. In other words: Denmark is an expensive country to live in/visit!
Personally, I wouldn't bring the battery grip, but your milage may vary. I would, though, recommend considering a lens in the 40-70mm "hole" you have. My 17-40mm used to be my walk-around lens on my 10D/50D, but after I changed to FF (5D3), the 24-70mm gets that honor. Again, YMMV.
If you're in need for a photo shop, I'd recommend Photografica
. They're a pro shop and actually know what they're selling, although not cheap. However, you can get the 25% VAT returned when you leave - ask about Global Blue.
As for Copenhagen itself, the city is quite low-rise, but with a lot of small towers (all different) - get up to the platform at the top of the Round Tower (on Kobmagergade in the pedestrian area) and you'll have a very nice view over the City. Likewise "Vor Frue Kirke" (Church of Our Lady) on Christianshavn has an exterior Ascent that will give another great view. There are plenty of buildings all around that are 100+ years old.
Frederiksberg Gardens and the Kings Garden (in central CPH) are favourites for the locals too.
If the weather is nice (ie: not raining), take the canal tourist tour - takes about 1-1½hour, and you'll see Copenhagen from a very beautiful angle.
Christiania is another classic, but be careful about the camera in Pusher Street.
As for wildlife, Dyrehaven (The Deer gardens) in Klampenborg is usually the place to go. It's usually invaded by photogs from all over Europe around September when the deer go into heat. Some have carts for all their "big whites".
For landscape, do consider that Denmark is as flat as a pancake.
Feel free to ask more detailed questions.
Oh btw, you chose a very good month for visiting Denmark: August usually has the best weather (most stable & warm), but bring a light raincoat - do not use an umbrella, as it usually rains either sideways or horizontal.