Thank you for posting the video, Queen Claude. A rainy, windy day in Denmark. Everyone looks good, as usual. Beneditke's outfit is quite autumnal.
A couple of questions: I recognize the Speaker of the Parliament. If the Prime Minister were to die in office would she take his position, or would there be a new election? In the US, we have a 'line of succession' if the President should die in office. (VP, House Speaker, Pres. pro- tem of the Senate.) That person holds the position until the next regularly scheduled election.
The male voice on the loudspeaker - was that a protester? I saw some signs, but I don't know if they were protests or not.
If the PM dies, another member of the government will become Prime Minister. I can't remember who at present, but that person has already been named and been appointed the vice-prime minister. That's not an official title as such though.
If there is no vice-PM appointed, the second highest minister will take over, that's traditionally the Foreign Minister followed by the Minister for Finance.
What happens in case the PM dies is that the vice-PM takes over temporarily. Once he/she has met with the government he will request an audience with the Monarch. Here he will inform her of the position of the government. QMII will ask him to recommend a new Prime Minister. He will then inform her that he has a mandate behind him to take over the government and QMII will then request him to take over as the new Prime Minister.
Having accepted that responsibility the newly appointed PM will then recommend whomever to replace him as minister, so that he can focus on being the PM. - That may happen at the same time.
The new PM will hand in his resignation as say Foreign Minister, which QMII will approve. - Then he will request QMII to approve of the new Foreign Minister, which she will. She will also meet and congratulate the new Foreign Minister.
When that's over, there will most likely be a presentation for the press in front of Amalienborg, before an official hand-over in the Foreign Ministry. Then the new PM will go on to Statsministeriet (The Ministry of State = The PM's office) and formally take over there.
It does not involve the Parliament or the Chairman of the Parliament, unless the government for whatever reason fragments of resigns.
It's an entirely different matter with the head of state, i.e. the Monarch. There must always be a head of state (or a Regent or a Rigsforstander) on Danish soil. (*) Which is why a member of the DRF is always in DK at any given time.
Yes, there was a minor demonstration, which is of course a democratic right - provided they have a permission from the police. Otherwise it's an illegal gathering and as such a public disturbance. But getting such a permission is usually a formality. - The police simply needs to know where the demonstration will take and where if any, route the demonstrators will walk. So that the police can guide the traffic and
protect the demonstrators.
I don't know what they were demonstrating against and no one seems to care.
A demonstration can be denied. You will for example
not get a permission to demonstrate at a funeral or near a funeral procession in DK. A reverend Phelps would get a big no from the police should he wish to demonstrate at a funeral in DK, and be arrested if he went ahead anyway.
It also happens that the police outlines the march route for controversial demonstrations, in order to maintain order and protect the demonstrators. Some 15 years ago, the police directed that the march route for a neo-Nazi demonstration should end at a garbage dump...
- Democracy and the right to demonstrate can be interpreted in many ways...
(*) Unless the Monarch is only away for a few hours.
Oh joy, oh joy, Daily Mail has covered today's event:
https://www.dailymail.co.uk/femail/...-joins-Princess-Marie-opening-parliament.html
Thank you, Muhler. I knew I could count on a good explanation from you! 
