"At the start of Phase 6 all Private Companies are nationalised into the Russian State Railway"

If a Public or Minor company is the owner of a Private Company, does the Private moves to the RSR?

Yes.

Yes, the RSR takes all of the private companies and pays their owners (players or companys) for them. Normally, by phase 6, all private companies will be owned by public or minor companies. Put the private company on the RSR, then take the appropriate amount from the bank and put it in the company treasury.

Note that at the start of phase 6, minor companies without trains that have operated at least once are nationalized BEFORE the private companies are absorbed by the RSR. The order is important - otherwise the RSR could get both the minor company and its face value in cash (for trainless minors owning a private company).

This is incorrect. The game designer (Ian) has clearly stated in response to questions that the nationalization order is the same as operating order. This does sometimes result in the RSR getting a double bonus. There was a lot of corruption in 19th century Russia, after all.