Disclaimer: Not a nurse and don't know much about nurses so just giving you my generel impression.
Are nurses well paid?
I guess this depends who you ask. A quick search tells me that a freshly educated nurse can expect roughly 4400 USD a month. This is not a bad salary in Denmark. "Well paid"... Hm not well enough if you ask me. It's not bad, could be better.
Are nursing unions strong enough to fight for better conditions for their workers?
Usually, or at least historically, unions have been very strong and effective in Denmark which has lead to generally high salaries. But yes as you note, unfortunately the government stopped a strike by law.
I think on one side there is a political problem in that the politicians did this, which as you note goes against the spirit of unions.
On the other hand, we need to face the reality that there will be more and more elderly people and less and less people to pay for their care. I'm not sure how to fix this but perhaps giving nurses too high of a salary is simply unsustainable. But I do think we can do better.
has there been a nursing exodus to other countries
I believe even before the strike, many nurses took substitute jobs in Norway because it paid better.
Where I am we’re free to call in sick (so far)
Everyone in Denmark is free to call in sick as well. You also usually get the first sick day of your child off.
are nurses in Denmark presented as people who should be happy to have the ‘honor’ of working helping people, meaning pay is not important?
I have never heard of such an attitude.