Facebook is the way to switch between machines, you don't ever have to log onto Facebook itself (other than to set up the account) to use it as a login. You might be able to transfer a portable's team to a FB that has no team attached to it, ask Nordeus Support

The "Let Me Play" is (and has been since mobile version available) tied to the device it is launched on. Maybe with your and others input they could change that, but I wonder how? FB is one of the more universal signins, what else would they use?