As a green card holder she will be liable for U.S. income taxes on her worldwide income, and have to file U.S. tax returns as a resident. The rules for spending time in the U.S. are not fixed on a certain number of days per year, but any trip spent outside the U.S. of 6 months or more calls into question whether she really is still a resident of the U.S. In many such situations, a person is better off with a visitor's visa (though in that case one also needs to avoid spending too much time in the U.S.).
As a permanent resident, your mother's income will be subject to US taxation but there are exemptions. You should check with a CPA or tax attorney regarding this issue.
As for keeping her green card, she must reside permanently in the US to keep her status. Only visiting will not be enough but there are ways to keep her flexible for some time by applying for a re-entry permit. Details should be discussed with an immigration attorney.