MB4 254b: Kutim, Heretics, Sinners and Eruv Chatzeros (385:2-4)

Kutim have the status of an אינו יהודי and one needs to purchase rights from them for an eruv chatzerot. What about a mumar or מחלל שבת בפרהסיה?

If one only violates איסורי שבת דרבנן does he have the status of a מומר?

What if someone only violates Shabbos in private?

Eruv chatzerot when Jews dwell with an apostate.