Just to have understood it correctly: If I have my XEM on my address for a long time, then I don't have to do the Opt-In and I don't have to be active for the snapshot either, right? Will the public and private key on the symbol blockchain be identical to those on the NEM blockchain?

I think that it is not.
The account must be attached to the "opt-in" process; e.g. in the NEM wallet will be a method (command or something) and by using that method you tell to the XYM system that you want XYM coins.