You clear immigration when you arrive in CPH and that will be the only time you’ll go through immigration (as you’ll enter the Schengen-zone). Your next official passport control will be when you return to the US. You will clear customs in all countries but the only which “matter” will be OSL as Norway is outside the EU customs union. There’re limitations on alcohol, tobacco etc which you can bring into the country.

In all places you’ll be from plane to curb in less than 30min, most likely a lot faster (assuming you have checked-in luggage).

You can very easily look up how long it takes to get into the city. Public transport is recommended in the Nordics unless you’re staying far away from the stations - though if three with luggage would recommend you get taxi’s. Oslo Taxi in Oslo, Stockholm Taxi in Stockholm are really only ones you should use.