What is the easiest way to transfer everything I can, especially passwords.

Opera should import your data. Just install Opera 12.17 before and make it the default browser.

Whick folder EXACTLY I hae to back up? And which files to copy in which new folder on new PC (file names are still the same??)

Note: Opera Blink (Opera 15+) is not compatible with Opera Presto, so copy the files won't help.

If you want to keep using Opera Presto, backup your profile folder. See Help > About for the path. Then install Opera Presto on the new pc and copy the files to the new profile folder.