Bergen Norway to St. Petersburg by plane

Does not look like: https://avinor.no/en/fly-fra-bergen/

Does not appear there are any regular commercial flights that do so.

In case you didn't know, if you fly to Helsinki, the fast train from there to St. Pet is now only 3.5 hours:

https://www.vr.fi/cs/vr/en/train-timetables-finland-russia?cmp=etusivun%20alabannerit&itm=Timetables%20between%20Finland%20and%20Russia

No, you can't. The fastest connection is probably SAS via Stockholm.

Here's a link for info on getting a visa to go to Russia, just in case you haven't explored that bit yet.