Looks like dost changes gender depending on a female or male friend, yes?
Yep! Exactly
Would ki become ka if we were talking about a male? Or is dost just a feminine word?
Yes. Eg: आमिर जूलिया का दोस्त है
Actually, दोस्त is a masculine word. It's use as a feminine noun like in the original sentence is not exactly grammatical (although it's commonly done).