Scandinavian trip.

Just a few partial answers:

I use Google Flights to search schedules and airplanes for transatlantic flights, then I book directly with the airline. Don't be tempted by third parties like CheapOair.

Tell us where you're flying from and others may have suggestions for budget airlines that might work for you, like Icelandair or Norse Atlantic Airways.

Your destination countries are very expensive (especially Norway), so it will be important to have solid information on your travel options. Rick's guide to Scandinavia will be helpful. He suggests penny-pinching techniques that you may want to take advantage of from time to time. (As for me, I bought a lot of cheese, nuts, bananas and Wasa crispbread at supermarkets in Norway and Sweden.)

If you fly into or out of Oslo, the regular train from the airport will be your cheapest option. It also allows you to use city transit from the train station to the area of your hotel if you need to. I think that's good for 90 or 120 minutes after you buy the train ticket, which you can do right at the airport.

I don't know the best website to use for hostels; you can probably find that easily by Googling. I've notice that booking.com lists some hostels, so you might also look there. Do be careful about locations. One way hostels keep their rates low is by (often) being located in less-than-central parts of town. You are planning to move very fast, and it would be best not to be staying a 45-minute bus ride from the center of a city.

You can do the Norway-in-a-Nutshell trip--at least in the Oslo-Bergen direction--without spending a night en route, but it will be a long day. I spent a couple of nights at the hostel in Flam so I could see the Borgund stave church. I had a private room with my own en suite bathroom. I imagine they have a room with three beds in it. Furnishings are simple, but the place is clean and well-located. Like everything else in Flam, it books up very early. If you want to break the NiN trip midway (edited to add: I meant without spending extra time in the area; if for more than one night, it's worth checking out Balestrand), Flam is the best place to do it, and you should book a room at the hostel ASAP.

Train tickets between major cities and ferry tickets are both likely to be cheaper if booked early. Be very careful about cancellation/change policies.

Others will be more helpful about your routing, but I will point out that Bergen has good airline service, so flying between that city and Denmark or Sweden (in either direction ) could be efficient.

While I understand all too well the cost of traveling in Scandinavia, I think it's a real pity to have only one day in Copenhagen, Oslo and Stockholm. An extra day would be nice in Bergen as well. I'd urge you to try to add a few days to your itinerary so you can have a better look at the major cities, especially Stockholm and Copenhagen.

I would like to stay 2 nights in each city of Oslo, Stockholm,
Copenhagen.

That's a very short stay in those cities, if you can add a bit of more time you will not regret it.

I already know I want to bike the cities--we are all bicycle riders,
or do a Hop On Hop Off bus/ boat and realize with just two nights we
will just get a general idea of each city.

All cities are easy to explore by bike, although parts of Stockholm and Oslo are a bit hilly. The public transportation in all cities is great though and metro, buses, trams and ferries are also good options.

Perhaps one museum and one thing the kids might like.

What do they like? For Stockholm, the Vasa museum and Skansen can be a good combination for a day.

1. best priced flights? Into Copenhagen? Stockholm?

Who knows? Predicting flight costs is tricky and your guess is as good as mine. But Copenhagen and Stockholm are the largest airports in Scandinavia and are well connected. Between the US and Scandinavia, SAS, www.flysas.com is often a good option. They have direct flights from Chicago to both Stockholm and Copenhagen.

Best places to stay there for four? Close to airport/station.

Stay in the cities, do not stay around the airports.

1. Best route? Copenhagen to Aalborg to Bergen to Oslo to Stockholm to Halmstad to Copenhagen and home?

That's not a bad option, although you can also do it anticlockwise or start in Stockholm.

How do i get from Aalborg, Denmark to Bergen?

Train to Hirtshals and then ferry to Bergen.

Should I do Sweden first, then Oslo and Noway in Nutshell and Bergen,
then to Aalborg and back to Copenhagen?

Maybe, there is really no right or wrong answer to that question.

1. Anyone have a estimated cost for flights? rail? most hotels and hostels?

Trains should be booked in advance for longer trips, they can get very expensive if you try to buy tickets last minute. For hostels in Sweden, STF is a good start, https://www.swedishtouristassociation.com. A memorable place to stay in Stockholm can be Af Chapman, https://www.swedishtouristassociation.com/facilities/stf-stockholm-af-chapman-skeppsholmen-hostel/

1. Who can help custom this trip and advise on travel insurance?

This forum is not a bad option.

ORDTraveler, the part of the Oslo-Bergen train trip between Oslo and Myrdal is considered more scenic that the section farther west. By just staying on the train rather than dipping down to the Sognefjord, you skipped 3 of the 4 scenic components of the Norway in a Nutshell route; that's probably why it didn't meet your expectations.

I agree with you that skipping Norway on this trip is worth considering since so little time is available.