بيتُكِ
Should be بيتُكِ
Is there a difference between "بَيْتِك" and "بَيْتَك"? I'm guessing one is used when talking to men and the other is talking to women?
You are correct. However, the diacritics of the two words in this context should be like this:
Masculine - بَيْتُكَ , Feminine - بَيْتُكِ.