You are certainly not rude for wishing to learn more.
Payment as in salary on an hourly/weekly/monthly basis like all other employees.
A LiW will get compensation for expenses, if she has purchase something she would need in regards to her duty as a LiW.
The same goes for personal items or toilet articles she is buying on behalf of the royal lady she is serving. - Neither QMII nor Mary can pop down to the local chemist, nor can they be sure of discretion if they were to order from Amalienborg. But letting you LiW buy or order it for you, then no one can be sure the article is actually for you. The alternative would be to let your chambermaid buy it for you. Better to let a trusted LiW do it, wouldn't you say?
Alexandra fell victim to such an indescretion years ago. She and Joachim's driver was sacked and as a private employee and not a court employee, he was not covered by a letter of confidence, so he told among other things about some very intimate articles he bought at a chemist to the papers. I believe it was Ekstra Bladet that happily printed every detail.
If a court employee breach the letter of confidence it's a criminal offence not just a matter for a civil lawsuit.
I've signed several such letters myself in connection with some jobs I've had.
I must admit I'm not completely familiar with the insurance issues, but if someone is injured in my house while being there as a guest or working for me, then it's my insurance that will deal with it, unless we are talking about an outside contractor. I guess that must apply to LiWs as well.
As for healthcare: If a LiW is not employed then a healthcare agreement for employees does not apply. Not all workplaces has such an agreement. She would instead be covered by the public healthcare system (and probably a private one as well) like everybody else.
ADDED: More of "hofstat". A Hofstat, I don't know the English word, is an organisation that along with other hofstater (plural) constitute the entire court as a whole. QMII has a hofstat, consisting of her closest associates. Frederik and Joachim each have a similar hofstat of their own. The royal garages is also a hofstat, and so on. All of these hofstater constitute the entire court working for the DRF. All headed by a boss i.e Queen Margrethe.