It sounds like you've but a lot of thought into this, and I'm very glad you have. Without reading the book, I can't know for sure what (if any) problems it may have. Getting a sensitivity read is a great step. A sensitivity reader is more than just a person of colour reading the book, but rather a person from a specific identity who is experienced in identifying racial microagressions and problematic representation associated with that identity. This is why I suggested a black sensitivity reader. There are a couple of things I should clarify about my comment above.
1- I did not say that white writers do not have to tools to write about racism, what I said is that they don't have the life experience to write about it with the nuance that a marginalized person could. And I do not feel that living among people of colour gives people that experience, either. There are reasons that publishing has recently recognized the importance of own voices for these stories. It doesn't sound like your story is about racism, though, so it's not an issue here. What is worrying me is that you believe growing up in area with predominately people of colour gives you the ability to fully understand what it's like. Just like I will never really and completely understand what it's like to have white privilege, you must admit that you can't fully grasp the experiences of those who have always been marginalised because of their skin colour.
2- it sounds like your Indian representation is well-done (but again, I can't know that without reading it). I am Indian, and the only advice I have for you there is to watch for stereotypes with the family, but I LOVE that you've chosen to have the best friend character Indian (we need more in fantasy!)
3- The biggest issue I see with with your black representation is the slavery. No matter the reason, 4 black people as slaves is a problem. Is there any reason why you can't change the race of the slave clan? Also, the part you spoke of when the lighter-skinned people covered themselves with mud to darken their skin is extremely troublesome, as it reminds me of black-face. And using dirt or mud to make a white person more like a darker person is a problem, no matter how you phrase it. The trope/misrepresentation of darker-skinned people being dirty or unclean is very old, and very prevalent, and it's the last thing you want to do is remind readers of. I have experienced racism first hand where people have asked me if I was really brown, or just hadn't showered in a while.
I hope this helps you! And good luck with your story!