Here is what the first (democratic) Constitution of 1849 has to say in regards to Regent/Rigsforstander.
It has to be said that back then monarchs didn't travel as much as now and with modern forms of communication and transport it's now very easy to get in touch with the monarch, when he is away from his realm. The Danish monarch had much more political power back then as well.
Rather than translating the text to "modern", I'll interprete and explain it instead.
§ 9. If the king, due to illness or because he is away, find that a Rigsforstander should be appointed. He will convene the Parliament and propose a bill. - That is no longer necessary. QMII just sign a document statiing that X will be Rigsforstander until she returns and then informs the government.
§10. If the king is unable to rule, the State Council (the government) will convene the Parliament. When the total Parliament, with 75 % of the votes that is casted, find it necessary it will appoint a Rigsforstander and if necessary a Guardian. - The Rigsforstander rules the country on behalf of the king, the Guardian look after the king's interests.
§11. Is there reason to fear that the Heir upon the death of the king will not be of legal age or for another reason unable to rule himself, a law will be passed which appoint a Rigsforstander and a guardianship is appointed by the king. - If the heir is under the age of 18, comatose, in rehab because he's a drugaddict, in captivity and so on.
§12. The Rigsforstander take the appropriate pledge/oath before the king and excerzise as long as he is Rigsforstander all rights by the monarch in the name of the king. He cannot propose changes in the order of succession however. - Nowadays they sign a pledge to obey the Constitution.
§13. When the king is dead, the Parliament will convene on the fourteenth day. - This is achaic legislation and presumably the Parliament approve that the succession is correct and that no coup d'etat has taken place.
§14. If there is no heir or the heir cannot lead the government at once, the State Council will head the government until the Parliament has decided what to do.
- The monarch is no longer head of the government. QMII is head of the State Council. The government is the udøvende magt = the executive power and act formally in the name of Queen Margrethe. The PM will run the store until the Parliament has decided what to do. Interestingly the first or one of the first things a newly appointed Rigsforstander has to do, is to approve himself.
§15. Is the heir or Rigsforstander absent, the convened Parliament will state a timeperiod in which he must return.
If the heir is not of legal age or for another reason unable to rule. And no Rigsforstander and Guardianship has been appointed, the convened Parliament appoints a Rigsforstander and authorise a Guardianship.
If there is no heir, the convened Parliament will elect a king and determine the future Order of Succession.
- Lets break this up in three parts:
A) Is used if Frederik is sitting in Australia with his arms crossed and sulking, because "I will not return home to be king"! Or if he is abducted by aliens or lost at sea.
B) Is used if Frederik and QMII dies before Christian has turned 18, or if he is mentally or physically unfit to become king.
C) If there is no (immediate) successor at all. - There is no "King Ralph" scenario here.
§ 9. Saafremt Kongen, enten formedelst Bortreise eller Svaghed, finder, at der bør udnævnes en Rigsforstander, sammenkalder han Rigsdagen og forelægger den et Lovforslag herom.
§ 10. Bliver Kongen ude af Stand til at regjere, sammenkalder Statsraadet Rigsdagen. Naar da den forenede Rigsdag med tre Fjerdedele af de afgivne Stemmer erkjender Nødvendigheden, udnævner den en Rigsforstander og anordner, om fornødent gjøres, et Formynderskab.
§ 11. Er der Anledning til at frygte for, at Thronfølgeren ved Kongens Død vil være umyndig eller af anden Grund ude af Stand til selv at regjere, bestemmes ved Lov en Rigsforstander, og et Formynderskab anordnes af Kongen. Rigsforstanderen kan ikke deeltage i Formynderskabet.
§ 12. Rigsforstanderen aflægger den for Kongen foreskrevne Eed og udøver, saalænge Rigsforstanderskabet varer, i Kongens Navn alle dennes Rettigheder; dog kan han ikke foreslaae Forandring af Arvefølgen.
§ 13. Naar Kongen er død, sammentræder 14de Dagen derefter uden Sammenkaldelse den sidstvalgte Rigsdag.
§ 14. Er der ingen Thronfølger, eller kan Thronfølgeren eller Rigsforstanderen ikke strax tiltræde Regjeringen, føres den af Statsraadet, indtil fornøden Bestemmelse er tagen af Rigsdagen.
§ 15. Er Thronfølgeren eller Rigsforstanderen fraværende, bestemmer den forenede Rigsdag, inden hvilken Tid han har at vende tilbage. Er Thronfølgeren umyndig, eller af anden Grund ikke i Stand til at regjere, uden at Rigsforstander og Formynderskab er bestemt, udnævner den forenede Rigsdag Rigsforstanderen og beskikker Formynderskabet. Er der ingen Thronfølger, vælger den forenede Rigsdag en Konge og fastsætter den fremtidige Arvefølge.