"Julia goes to her sister's home."
Translation:जूलिया अपनी बहन के घर जाती है।
Apparently it is because there is an (invisible) "ko", which means "to" (Julia goes TO her sister's home). As there is an (invisible) postposition here, the words just prior to it convert to the oblique form, hence instead of "ka ghar" it will be "ke ghar", signaling the invisible ko (which my hindi book refers to as a "ghostposition"). You are supposed to use the oblique form before any postposition, and in this case before any "ghostposition" too.
So the difference between ke ghar and ko ghar is not noun gender, but case?
apni means the sister is Julia's sister. Uski would mean the sister is someone else's sister, not Julia's. Use uski if you want the sentence "Julia goes to his sister's house." In English the 'her' is ambiguous: is referring to Julia as the 'her' or some other female as the 'her'. In Hindi in this instance this particular ambiguity doesn't exist.