OP, welcome to the forum, and well done for doing your research about the realities of the move. Good to ask questions here, about how things work here. I'm posting this while I know nothing at all about the way things work in Brazil. This is just about Switzerland.
Tax
Here, a married couple is regarded as one tax unit. Therefore, for the tax here, all of your earnings (in Switzerland and worldwide) plus all of your husband's earnings (in Switzerland and worldwide) must be declared. Your joint tax will be calculated on that total sum. This includes all parts of your and his income generated in Brazil.
You'd have to consult the agreement, linked to above, about preventing double taxation, to see which of the two countries might exempt part of your income from tax because it was already taxed in the other country.
It will be relevant to your and your husband's tax situation whether your husband's business is a separate legal entity from him as a person. That might be a good route to follow, so that the business stays in Brazil even though the man is moving to Switzerland.
Permits and permission to work
If your work is secure and you have a permit, that's good for you. You will probably need to apply for permission to bring your husband and child(ren) with you. This is not automatically granted.
The reason for this is that the Swiss government is carefully trying to avoid letting people into the country who will get into financial difficulties such that the state will have to support or repatriate them.
You will need to demonstrate that you earn enough to support your husband and child(ren), and that you have large enough accommodation for each person to have some space. For example, a one-roomed apartment is not considered okay for a family.
If your husband gets a permit to live here, it will specify whether or not he is allowed to work here, too.
He cannot be classified as "unemployed" because that status is only for people who have worked here and contributed to the mandatory unemployment insurance for over a year or two. Accordingly, he will not be eligible for unemployment benefits.
For help in applying for a permit for him and your child(ren), I second with the suggesiton above that you ask at the support office for staff and students, at your Swiss university, as they may very well be used to these kinds of questions.
Child allowance and care
There's a small amount paid each month, on top of your salary, per child. It is a few hundred Francs. As far as I know, this is simply paid to all people legally living and working in Switzerland.
Child-care is a separate issue. Children are often cared for fully by one parent or by both parents each working part-time. There are some government and many private child-care institutions. Their fees are high. Please see the other threads on this forum, also about schools, if your child(ren) are of that age.
Good luck as you plan things, and soon welcome to Switzerland!