The die in this sentence is a relative pronoun. It means "who," and it refers back to Freuen. To help you understand, I'll give you the exact word-for-word translation.
Ihr seid starke, unabhängige, schwarze Frauen, die keinen Mann brauchen.
You (y'all) are strong, independent, black women, who no man need.
no article -> strong inflection
It isn't. Sie sind starken Frauen does not work, because the adjective declension for plurals in nominative case needs to end in -e.
Sie sind starke Frauen would mean "You are strong women" (polite, to a group), or "They are strong women". The whole sentence is in nominative case.
...starken Frauen could be dative case (either strong or weak inflection), so it would appear in sentences like Wir wünschen den starken Frauen Glück ("We wish the strong women luck").
Sure, just be aware that the use of "y'all" is basically confined to certain dialects (not all English-speakers around the world use it), and so you might find examples in Duolingo where it is not accepted (but if you're determined to promote its use, you can report it each time you find it).
And I do just that each time. Even if it´s been a year and it's still marked wrong. It's hard to change how you look at ways of writing when you've spoken like this your entire life. I wonder though; if southern Germany has any dialect similar to southern American English.
There are definitely dialects in German - many even stronger than the equivalent of southern-US English. But all Germans essentially learn two languages these days: standard 'Hochdeutsch' (in school) and their local dialect (in conversation). So Germans are well-accustomed to remind themselves of what is and is not a part of their dialect when speaking to others. As an Australian, I also do the same when speaking to others - but it was being exposed to 'foreigners' (and becoming one myself) that trained this response in me.