I want to protect a pdf in such a way that when people want to print it, it is allowed but on the printout a sentence stating that "this is a copy and is only valid on this date (the date of today is inserted)".
Yes, you can add a form field that only prints, but is not visible on the screen. This is one of the field settings. A script will be needed to set the date. This will only work when the form is used in Acrobat or Adobe Reader.