+1 for the Iceland layover.
For weather, best time to visit is either in the middle of winter, or in the middle of summer. Don't believe this crap about how amazing the nature is during fall and spring, it's not. You'll be miserable in endless rain then. Go either Nov-Feb or late June-late August.
Denmark is very different from other Nordic countries. It's very flat and there isn't much to do apart from visiting Copenhagen, which is a cool city, and if you have kids, the original Legoland in Billund.
Norway is beautiful, especially when driving the coast up north. I'd recommend renting a car, and driving. You see the most that way. I would avoid the cruises etc that they do to the fjords, they are generally not good value (and everything in Norway is bats*it crazy expensive).
The few Swedes here will perhaps argue otherwise, but I'd say the only thing worth seeing in Sweden is Stockholm, which is a very cool city. Perhaps my favorite city in all of the Nordics.
Visiting the region without visiting Lapland is almost a crime. The best infrastructure is in Finnish Lapland (that is, easiest to get in/out, good amount of hotels and so on) In the summertime you'll get the midnight sun (it's quite weird when it's 2AM and the sun is up), if you're into hiking/nature/other outdoorsy stuff, there's tons of that there. The earlier in the summer you go there, the less mosquitoes they have. They are annoying, but don't spread any diseases.
Oh, and the only place to experience _real_ sauna is in Finland. The Swedes/Norwegians have no clue how to do it properly.
Even though as a concept it sounds weird and painful, it is very relaxing. I'm ashamed to admit my wife (American) outsaunas me (a Finn) nowadays.
Are you flying around or driving? Driving would perhaps be a bit boring for the kids, but you see much more that way. Driving, you could fly in/out of Copenhagen, drive across to Sweden, drive up to Oslo then all the way to Nordkapp following the coast (very pretty road all the way from Trondheim), then drive down to Finland, could stop for some trout/salmon fishing in the Teno river, drive through the Finnish "lake district" (Finland has almost 190'000 lakes) down to Helsinki. While in Helsinki, you can do a day-trip to Tallinn, Estonia. It's very different from any Nordic cities you will visit. I wouldn't stay overnight there, just take the first ferry in the morning (it's a 1,5hr boat trip from Helsinki) and the last one back. You can see the Tallinn Old Town in a day easily. Their City Hall is from the 13th century, alot of old interesting architecture in Tallinn, something you won't get that easily in the Nordics.
Then take a cruise ferry across to Stockholm, then drive down to Copenhagen (6 hour boring drive) and return car back there. This way you would see pretty much everything the Nordics have to offer. Easily doable in 2-3 weeks.
Traveling between countries is easy, there are no customs/border controls, you just pass a sign that says you are now in a new country.
Flying would limit your options on spur-of-the-moment sidetrips, Flying internally within Scandinavia can be pretty expensive (or cheap, if Norwegian Air flies between the cities you want).
