Work permit and autorization of stay are different things, at least in Luxembourg and i believe the same apply to Netherlands.
I'm from Brazil and recently moved to Luxembourg. I received a work permit and a residence permit (i really don't know how to call this one, it's in French, "autorisation de séjour"). My wife only received a residence permit that is valid as long as I stay here. She is not allowed to work here. To do this, it's another process. I think it would be the same to you.
Brazil also has tax agreement with Luxembourg. This means that my "pension" deductions in Luxembourg will be valid in Brazil. As a practical example, I can count my work years from here for the government retirement in Brazil, when I come back.
I don't know if your wife will work here or receive a grant from the university. If so, she will be deducted by local taxes, what will also give you (both) access to public health there. As you don't have a local salary there's no tax for you.
Last, if my wife worked remotely from Luxembourg and received a salary in Brazil, in a brazilian bank account, this would make no difference to taxes here, as far as I know. One thing to note, we have to declare in Brazil that we are living abroad. This changes the type of annual taxes declaration (in Brazil) that we are used to do. If she worked remote, i don't know how we should proceed on this. I will check more on this topic and come back.