Hello. If I buy the full version, can I continue from the demo or do I have to replay those levels again? Thanks.

I don't think so. Or at least for me it made me redo the demo levels.

Actually you can, by entering "lonerider" (without the quotes) while in the Options screen, iirc.
This will unlock the very first tutorial mission, as well as the second "Running late on payday" and the third "Troublemakers in Flagstone". Won't carry over any of your unlocked badges though, but it should let you access mission 4 "Until death do us part" right away, I think.