Short and simple said the "value" of a bill just exists because of peoples trust in this value.
To be honest I do not really understand what you mean with your first question: "Can the Government use to print any value to their currencies?"
To the printing many bills to face debts question: that is exactly what is happening in Venezuela at the moment. Their economy crashes and many people are so poor they can't even buy themselves food. Usually it is or at least should not be possible for the government to print money as they wish. Theoretically (!!!) the government has no power concerning how much money is printed but yeah if the just do it as it has happened in Venezuela the HUGE negative effect of it is hyperinflation (the fact that Venezuelas leadership didn't thought about that or if they thought about it but still went on is just so bizarre).