I'm not a huge fan of southern accents, but that is just me personally. I'm from northeast United States (New Jersey) but I visit North Carolina frequently for family and they all have very strong accents. and no, I don't think that's at all strange!
(Note I'm assuming you mean southern US accents specifically, since many countries have "southern" accents of different kinds)
I don't really think much of them anymore since I live in Texas and hear them around quite often. Since it is common enough here, I have been desensitized to it I guess. I wouldn't find it odd for a mixed girl to have a country accent around either. I think it is less common to have than white people (from what I've seen), but it wouldn't strike me as odd.
Not at all. Many African Americans are Southern. Surprisingly, here in California, there is a group of descendents of "Okies" and "Arkies" who still somewhat have the pronounced Southern accent, who live from Stockton to Bakersfield.
I think accents basically come down to where one is raised and the culture one is raised in. A friend of mine thinks southern accents are sexy. I think they are okay. Hard accents can be difficult to interpret. I don't think it's far-fetched for a mixed girl to have a strong country accent.