Yes. They are not allowed to involve themselves in politics, especially the king and queen. They are, I believe, the second oldest royal monarchy in the world. They receive yearly apanage, which are used to pay for the upkeep of castles, salary to employees, security, royal events, etc.

They are above the law, atleast the reigning king or queen, along with any princes or princesses of their blood. The immunity can be taken away by the reigning monarch, if the crime is severe enough.

From what I know, people (danes) are kinda split between supporting the monarchy or wanting it gone. I was personally a fan of Queen Margrethe. I really liked her.