My wife and I went there last year and loved it. Very similar itinerary. I think I posted a trip report. We found the Copenhagen card to be cost-effective - particularly because it covers transit as well. You're seeing a lot of sites - a five day pass would work great for you . The canal boat trip is great. We went on it too late in the day for the cheaper company (I forget the names). If you use your Copenhagen card to get the free trip, you have to "buy" the ticket at their office on Ved Standen. You don't get the discount from their departure point in Nyhavn. Other highlights, tips, and things that might interest you.
We woke up early one morning and nothing was open. We took the train to Kastellet park and just walked around (and, of course, saw the little mermaid nearby). St. Alban's and Geffion fountain nearby were beautiful. That ended up being one of our favorite mornings.
If you're going to Christiansborg palace in the morning of day 3 and don't mind a little splurge for lunch, make lunch reservations at Nyllist i Taarnet - the restaurant in the palace tower. It's beautiful, the food is good, and you get to skip the line for the elevator to the top.
Note that your Copenhagen card will pay for the train trip to and from Roskilde. If you go to the ship museum, consider pre-booking a a trip on one of the viking ships - it's only an hour or so, but it's fun.
We did Helsingor and Frederiksborg in one day together. We like Elsinore better than Rick Steves does. On the other hand, we wished we left more time to see the gardens at the latter. The little boat ride around the lake at Frederiksborg is included with your Copenhagen card. We used it to get from the gardens back to the nearby town.
Two other things that are worth seeing are Reffen (a massive assemblage of food carts and sometimes performers) and Tivoli Gardens. We didn't ride the rides at the latter, but it was worth going into just to walk around.
We stayed at Hotel Danmark (near Tivoli Gardens) - the room was tiny, but the staff was friendly and they had a wine hour just before dinner, and a nightcap hour. Both gave us a chance to interact with other people staying there.
We never went to Christianshavn, but some people like it (not our thing)
We did do the changing of the guard and it was very underwhelming. Unless a royal is in town (I think you can tell by the flag they fly), I'd skip it.
Our itinerary:
Day 1: Arrive and walk the city (incl. Round Tower) ending with a canal cruise. Dinner at Host (it's Michelin starred, but we were disappointed)
Day 2: Kastellet Park (last minute add), National Museum, and Christiansborg Castle. Dinner at Rufino Osteria (recommended)
Day 3: Frederiksborg / Kronborg (Elsinore). Dinner at Karla's (recommended)
Day 4: Roskilde - Viking Ship Museum and Cathedral. Dinner in Tivoli Gardens (way over-priced)
Day 5: Rosenborg Castle / Changing of the guard (underwhelming - skip it) / Reffen. Dinner at Iluka (again starred and again disappointing - I guess we're just not into those architectural meals).
Day 6: Laundry and shopping and walking about town. Lunch at Cafe Sorgenfri (really good). Dinner at Nr.30 (good, but pricey). We thought about going to Dragor this afternoon, but we were too tired for another side trip. Kinda wish we had.
Day 7: Leave for Norway.