Is keeping an original faith accepted today? For example: A Catholic princess of Belgium wanted to marry the Lutheran crown prince of Denmark. I'm not entirely certain, but I believe that there is a clause in the constitution that says members of the royal family must be Lutheran. If the princess did not want to give up her faith, would the marriage be allowed? Or, would parliament issue a special law (as they do for immediate citizenship of spouses of Danish royalty) for that princess if she agreed to raise her children in the Lutheran faith?
I believe the Danish constitution only says that the monarch must be Lutheran. In practice, however, all non-Lutherans who have married recently into the DRF have converted.
In Sweden, the Act of Succession says that both the monarch and all "princes (or princesses) of the royal house" must be members of the Lutheran Church of Sweden. A prince or princess who converts to another faith loses his/her place in the line of succession.
In the Netherlands, the royal house of Orange-Nassau is historically associated with the Dutch Reformed Church (now the "Protestant Church in the Netherlands", or PKN), which is Calvinist/Presbyterian, but there is no legal requirement that the king or members of the royal family be Protestant. In the past, a non-Protestant bride/groom could have trouble getting the consent of the Dutch parliament to marry a royal (it was certainly an issue when Princess Irene and Princess Christina were engaged). Nowadays affiliated Protestants are only 18 % or so of the Dutch population though (most people are secular/non-religious), and I don't think that is an issue anymore. Willem-Alexander married Maxima, who is Catholic and didn't convert, but Maxima agreed to get married and baptize their 3 daughters in the Protestant Church. W-A himself is said to be a committed Protestant, though not as openly as some of his ancestors. He has been criticized occasionally by the leadership of the PKN for "hiding" his religion in public (e.g. not using biblical references in his Christmas speeches etc.).
In the UK, the law says that, in order to be in the line of succession, a person must be a Protestant descendant of Sophia of Hanover. In addition, the monarch must be in communion with the Church of England (i.e. the Anglican/Episcopalian church). Until recently, marrying a Roman Catholic also constituted legal grounds for exclusion from the line of succession, but that provision has been now (retroactively) repealed by the Succession to the Crown Act 2013. Since princes and princesses of the United Kingdom could not marry Catholics (or else face exclusion from the succession), senior princes/princesses traditionally turned to other Protestant sovereign families to find suitable brides/grooms, usually from Germany, Scandinavia, or the Netherlands.
Spain historically had a Catholic monarchy. In fact, Spain was the leading European nation in the counter-reformation movement and the Catholic inquisition. The modern constitution of 1979, however, ended the status of Roman Catholicism as the established state religion and that there is no longer a requirement that the king or the royal family be Catholic. In practice, they are Catholic though and it is unlikely that will change any time soon. Queen Sofia, who was Greek Orthodox, converted to Catholicism after marrying Juan Carlos.
The Kingdom of Belgium on the other hand was born in a revolution, which, among other things, was against both absolute monarchy and Protestant rule. Ironically though, independent Belgium's first king, Leopold I, was a Protestant (Lutheran), who had been even married before to the heiress presumptive to the British throne. Leopold later married however the daughter of King Louis Philippe of the French and raised his children as Catholics. The Belgian Royal Family has been Catholic since then although, again, that is not required by law. Princess Astrid of Sweden, who was Lutheran, converted to Catholicism after marrying into the family.
In the old Kingdom of France, only Catholic male descendants in male line of Hugh Capet could ascend the throne. As mentioned before, Henri of Navarre, the first Bourbon king of France, converted from Calvinism to Catholicism to become King Henri IV.