Doesn’t being Catholic mean that one should accept the Church’s teachings without publicly disagreeing?

Your thinking is correct, and you are certainly not alone, although you may not often see other non-dissenting Catholics on mainstream television.

Regarding the issues the woman mentioned, John Paul II definitively ruled out the possibility that the Church had the authority to ordain women in Ordinatio Sacerdotalis. However much this lady might hope for the ordination of women, it won’t happen because the Church can’t ordain women. Men are the valid “matter” for the sacrament of holy orders, just as surely as water is for baptism and wine for the precious blood.

As for receiving communion in a Protestant church, Catholics believe that the Eucharist is a sign of unity. This is one of the reasons that Protestants can’t ordinarily receive Communion in Catholic churches. The same holds true in the opposite direction: For a Catholic to receive Protestant communion would not only give the impression that the Protestant version is valid, but it would also create a false sense of unity.

Beyond that, when you realize that non-Catholic churches don’t have a valid Eucharist, you have to wonder why any Catholic would want to receive communion in Protestant churches. Perhaps what this woman really lacks is an adequate appreciation of what we as Catholics have in the Blessed Sacrament.