Ki and ka are used depending on the person or object they are referring to. If the object or person is feminine, you use ki while if it's masculine, you use ka.

Example - Peter's sister - Peter ki bahan

Here we would always use "ki" because the word sister is feminine. So it doesn't matter whose sister we talk about, we will always use "ki".

Olivia's house - Olivia ka ghar

Here "ka" refers to ghar(house). Since ghar is masculine in Hindi, we will always use "ka".

So

Peter's house - Peter ka ghar

Sarah's house - Sarah ka ghar

You just have to remember that ki/ka is not being decided by the subject. The house(ghar) could belong to a male or female but we would always use "ka" because house(ghar) is considered masculine in Hindi.

Correct. beti is female, so it requires ki.

You use के when you talk about Raj's sons.

राज का बेटा = Raj's son
राज के बेटे = Raj's sons
राज की बेटी - Raj's daughter
राज की बेटियाँ - Raj's daughters