I am distrustful of the Khazar story because I would like to believe it.

As for what people look like, it takes very very little genetic drift to substantially change how people look. There was a 60 Minutes article about a group of black South Africans who claimed to be descended from Jewish immigrants of Solomonic times. Some genetic testing was done on them that tended to verify this. (Most of them are now practising Christians).

As to conversions, there is a tribe of Sudanese whose chief led them in converting to Judaism early in the 20th Century. They are known as the Abu Yudaya and some of their descendants still exist. I believe they have set some of the Hebrew liturgy into African chants and maybe there's enough material there to start a thread above the line.

I don't think the case for Khazars having any modern descendents has been made. But it's a great story.

Jews tend to believe that it's not what you know or who you know that gets you to God, but being true to your best self and how you treat your fellow humans. It is not necessary to be Jewish to attain your ultimate reward.
And if you want to be Jewish, you should have your head examined!