I can't tell you about the Danish monarch but in the UK the monarch is most a figurehead for practical purposes, although all sorts of things go through the monarchy.
After a general election the monarch invites the leader of the winning party to form a government. In theory the monarch could invite anyone they choose to form a government, elected or not. I believe in theory the monarch could pick up the phone and invite you or I to form a government. It's hard to imagine them actually doing it, although when an election results in a hung parliament (where no one party controls more than 50% of the House of Commons) and no credible alliance forms to create a majority, the monarch is arguably more likely to invite the leader of the largest party to form a government.
The Prime Minister meets with the monarch regularly, and bills passed by the Commons and Lords are sent to the palace for royal assent. In practise I believe this is little more than a rubber-stamping exercise although in theory the monarch could refuse to sign the bill.
Members of the Royal Family, at least the ones who aren't writing books griping about their life of extreme privilege, typically also act as figureheads for charities. Their royal seals ("by appointment to HRH The Prince of Wales" etc) that they may award to certain businesses indicate royal approval of the products that business sells, and so are highly regarded.
