There's actually two questions here, The first one is the easiest to answer: Once baptized into the Catholic Church, a person remains part of the Church for life unless that person leaves the Church through a formal act. So, yes, the woman in question would still be a Catholic. As for the second question, there's a lot of varibles that could affect the answer to this question. While this answer will address the most common cases, the best answer can be achieved by speaking with your local Catholic priest. Since 1983, the Code of Canon Law requires that Catholics be married before a Catholic priest (or deacon). Canon Law also requires that the marriage take place in a Catholic Church. These requirements, are, however, waiverable (called a dispensation). If the woman in question received such a dispensation, then her marriage is considered valid and has no negative effect on her standing with the Church. Even if a dispensation was not recieved at the time of the wedding, it is possible for the marriage to be convalidated (commonly called "blessed") by a priest after the fact. If such a convalidation has been received, then this woman's marriage is considered valid, and has no negative impact on her standing with the Church. If, however, no dispensation was granted and the marriage has never been convalidated, then the woman in question has not received the sacrament of marriage. If the woman is participating in maritial relations with her "husband", then she is committing either the sin of fornication or adultery (depending on the specifics). Both of these sins are mortal sins and would preclude her from receiving Holy Communion. If neither of the parties have a living ex-spouse, the situation is easy to resolve by getting the marriage convalidated and going to confession. The first step is to make an appointment with you local Catholic priest. If one (or both) of the of the parties do have a previous marriage, the situation is a bit more complicated, but the first step is still to make an appointment with your local Catholic priest.

As long as she does not renounce her Catholic faith and she brings the children up Catholic she can. 50 years ago it was different. The Catholic Church has become very tolerant. Even marrying a Muslim, Jew, Buddhist is accepted. General belief is that all people believe or do not believe according to upbringing, culture and understanding.