1) Is there any fast option how to move existing connections into folders without recreating them in that folders? I don't see any option like "Move to folder" or something...
That's true. You need to use drag&drop.
2) I see the way of switching among multiple connection has changed. Formerly there was a combo on the right side, now it's organized into tabs. Is there any setting to change this back?
No, the session drop down menu was discontinued.