Provided that the Catholic man did not formally leave the Church, and given the circumstances you described, the first marriage was not valid. But to determine that, he or she would have to apply for an annulment. One of the parties involved in the first marriage should speak to a priest to help get the process going, or contact the marriage tribunal of the local diocese directly. If it is determined that the marriage is null, then she could marry in the Church.
The Church has such high regard for marriage that whenever anything even looks like vows have been exchanged, the Church assumes validity unless proven otherwise.
In the meantime, provided the woman has not entered into a second marriage before the first is declared null, she should be able to enter the Church.
Your second point is well taken. Ideally, a Catholic should not enter into a relationship with a divorced person until an annulment is granted. Sometimes annulments are not granted, and then what? Once romantically involved, it is very difficult to break it off. This causes many people to enter into invalid marriages.