What time of year are you planning to go? The weather and the amount of daylight hours will have a big influence on what you can do and when.
Also be aware that there may be issues with small kids and public transportation or museums. For example, prams are not allowed on the special boats that go to the island near Oslo that has the Viking ship museum and other maritime museums. The obvious reason? Getting them on and off the boat is a serious struggle up and down stairs and they take up an enormous amount of space on board. You may be limited to bus or tram options.
Likewise, maybe they were there, but I noticed no prams or baby equipment of any kind in museums, castles, city halls or any other kinds of public facilities we visited on the RS Best of Scandinavia tour I was on last month. If anyone else has observed anything different, please correct my impression.
On the positive side, there are tons of great things for kids to do, especially outdoors. Again, that's highly dependent on the weather and hours of daylight.
I agree that you have too many locations for your number of days. Think in terms of nights, not days and keep in mind that your first night is on the plane, your first day will be your jet-lagged arrival day and your last day is on the plane home. Two nights are required to have one full day between.
Most people fly. Here's why.
The train ride from Copenhagen to Stockholm is at least 6 hours. From Copenhagen to Oslo is at least 9.5 hours. From Stockholm to Oslo cross country is at least 7 hours. Driving would probably take longer.
Helsinki is definitely the outlier. Of course there is no train option directly from Stockholm to Helsinki. By boat, then by train it is at least 14 hours.
You can use DB BAHN to check out these routes and see maps of them, although you can't buy tickets for them from DB BAHN.
I'm sure you're researching the cities you listed for kid-friendly opportunities, although most of what you find will probably be more appropriate for kids a few years older than yours. Based on that and your short timeframe, two locations would likely be the maximum, flying into one and back home from the other. Renting an apartment in those places might be the best option for your very young family.