While all of the characters all speak Danish in the film, the court language in Denmark at the time was actually German. In real life neither Count von Bernstorff nor Johann Struensee spoke any Danish, and it is more than likely that Christian and Caroline also conversed in German rather than the "people's language."
In 1769, when Queen Caroline Mathilde is taking tea with the Dowager Queen Juliane Marie, they are clearly seen to be drinking from Danish Flora Danica cups. The Royal Danish Porcelain Factory was not established until 1775 and the service was not put to use by the Danish Royal Family until 1803.
Juliane Marie tells the servants that if they know about the affair and try to conceal it, it's a mortal sin and they will end up in purgatory. The danish religious view in the 18th century was Lutheran protestantism, which denies the presence of purgatory.