You should always be available by phone, within reason. Off hours there should be a supervisor who should be completely competent to handle most situations. However, the supervisor should always get the feeling that you are available for your unit.
The issues off shift should be assessed and triaged like any situation. I do not think your staff (except maybe 2 trusted charge nurses/ANMs) should have your home number, they should always go through the supervisor. The supervisor should be given a chance to build a relationship with the staff and have a fair chance to resolve issues, and you should support their efforts and decisions. Remember, they cover for all NMs off shift and have to balance the workforce and workload of the entire facility. They are an extension of all of you.
If you feel the supervisor is reaching out to you for issues that either she should be able to resolve, or can wait, within reason, until a civilized hour, then you need to speak with the DON about her (or his) assessment skills and decision making process.
Ask who will be your mentor. You are right, the DON will not be the best mentor for a variety of reasons. Perhaps another NM would be a better match. Seek mentors out, whether they are ANMs, payroll, human resource managers etc.
Can I recommend a book to you? It is called the "Carrot Principle" and it will inspire you to lead and manage and has sustained me through some of the most challenging teams.
And yes, after 18 years as a clinician, I've spent the last 5 in management.