Here are some examples of women’s congregations that have gone canonical:
benedictinewomen.org/
mtabor.com/
In answer to your questions:
The Catholic church has not legally guarded the word “catholic” so it is used by all kinds of groups on the street. Ask any Anglican to explain his faith to you, and you will hear the word “catholic” within 30 seconds, guaranteed. Nevertheless, non-canonical groups tend to be a little bit careful about the word “catholic,” although I don’t think that dioceses can do much to them, to be honest. There is a huge amount of easy-to-invoke imagery that suggests Catholicism and that’s far more effective, as well as being much less risky to use. And the imagery is VERY VERY EFFECTIVE for the purposes of gaining goodwill with people in the neighborhood and maintaining funding.
Non-canonical means that an organization no longer has canonical status. Non-canonical status can be as simple as a break from the diocese. Once the dependence on canon law has been denied by the group, they no longer have the (somewhat dubious) advantages of being recognized by the Church, but they do retain almost all the advantages. If they have done their legal homework with respect to property law, yes, they can take all the property and be none the worse off for having gone non-canonical. The Church does not own property that belongs to congregations; the congregations do. They church can’t usually touch it.
Sisters belonging to these congregations usually have left the church. Although if you are out of the reach of canon law, it really has no impact on you and you don’t need to acknowledge any disciplines such as owning up to where you are spiritually or organizationally. You can do as you please as long as your lawyers keep you out of civil trouble. See the Benedictine Women of Madison for how that looks.