There's no work authorization that covers this sort of arrangement whether you're Danish or Canadian, so effectively you cannot legally do it. Basically the law hasn't caught up with technology.
There are convoluted ways of doing it, for example if your spouse has L-1 or E-1/2 then you can get work authorization in the derivative category as L-2 or E Spouse.
It's something that has always been really silly, for example say you are a travel writer who writes travel books. Is it illegal for you to start writing the book on your laptop while in the US and you have to wait until you get home? Is it even legal to enter the US because technically you are "working" rather than just visiting?
Basically a very grey area of the law, the State Dept. simply comes back with the stock answer that journalists must have an I visa, but go and check the shelf in your local bookstore and see how many of those books look like they were written by people with visas.
The other common example is say you were in California and someone phones you up from the office in
Denmark to ask you a question - well at that point you are technically "working". However caselaw is that "incidental" work like that does not violate visitor status.
I was in a situation once where I was using my laptop in a transit lounge in the US. At that point you haven't even legally entered the US, but yet I was "working" in the US.
On the tax situation, the answer is basically yes, you should pay
taxes while working in the US on your income. But it's hard to do in this situation as essentially your employer (or you) would need to set up a US payroll and put you on it. There are usually exemptions in the tax treaty for short-term work that is under a certain dollar limit, but you'd have to check the US-Denmark tax treaty.
Read IRS publication 519 and check the treaty provisions in publication 901.
It ain't the EU.

Steve.