The website https://www.flightsfrom.com/BGO?timeFrom=0&timeTo=1440&from=BGO&aircrafts=&state=1 indicates that Bergen has flights to all the other cities you mentioned, but perhaps just one flight a day to Paris. You can get more precise schedule information at www.skyscanner.com . Use dates from this fall that are close to when you expect to travel next year. Make a note of the fare pattern by day of the week, because that may save you some money when it's time to buy your tickets.
You can use the Danish (https://www.dsb.dk/en/), Norwegian (https://www.vy.no/en) and/or Swedish (https://sj.se/en -- corrected) rail websites for schedule and fare information. [See Badger's post below for his preferred Norwegian website.]
I believe you'll find it's not practical to travel to or from Bergen by train unless you want to stop off in Oslo--which most people do. It's very popular to travel between Oslo and Bergen via some form of the Norway in a Nutshell route. That allows you to see some spectacular fjord scenery as you cross southern Norway. It's easier to get to Oslo than to Bergen via ground transportation from Copenhagen or Stockholm.
Bergen is a very pretty city. I liked it a lot, but even I (one of the very slowest travelers on the forum) only spent 4 nights in Bergen. For most folks four nights wouldn't be necessary unless a fjord day trip was planned.
In terms of routing strategy, I like the idea of flying into or out of the place that's trickiest/slowest to reach by train or ferry--which would be Bergen. But a transatlantic flight into Bergrn may not be affordable or even possible (depending on airline allowances). I'd use Google Flights to check the trans-Atlantic flight situation.