Even though it is later revealed (https://www.imdb.com/title/tt0589874/) she never graduated high school and spent much of her life as a housewife before Charlie died. She also states in one later episode that she does not even believe in psychiatry. And yet, in the Christmas episode (http://www.christmastvhistory.com/2012/01/betty-white-golden-girls-christmas-1986.html), she is shown counseling patients.

Grief counselor is a generic title. There are basic trained people even on suicide hotlines that take basic information and schedule you with a professional. Just being a sounding board could be a "grief counselor." Who knows she could have met them, talked to them what was upsetting them and referred them to a professional therapist. Is there not an episode where Sophia makes a joke comment about Rose meeting with people in crisis or something?

Yep, she wouldn't be licensed but could just be the first person the distressed person talks to and then she refers them to a professional.