There are certain times when there is a surplus of energy in the network (usually at nights iirc).
With a lack of good storage mechanisms, energy vendors need to actually pay companies to use the energy at certain times. Otherwise, the energy excess would damage the network.
If Denmark doesn't have enough companies that could take the energy, they have to send the energy to be used up somewhere else. It's quite likely that at the same time in other countries a similar situation occurs, so nobody really wants the extra electricity. Hence Danes need to pay other countries to get rid of it.
Possibly, but they'd have to be designed to be remotely controlled (possibly at an individual level). They'd also have to be specifically engineered to withstand the sudden reduction in countertorque when the windmill is removed from the grid and then to lock the blades against windmilling while shutdown. And of course, to reverse the whole process on command.
I would assume the blades can be "feathered" so as to catch very little wind and not impart a rotational force, as on a prop-drive aircraft when one of the engines needs to be shut off. There has to be some way to take them out of services for e.g. maintenance.
What's surprising is there's no way to disengage the generator and just let the turbine spin for nothing when there's excess power. Seems they could push around seawater if they need the resistance for structural integrity.
