The first sentence is fine and is understandable in context. かれはむすこです。 perfectly fine. The second sentence, however is out of order. You should say: かのじょはむすこのむすめです。 To make it extra clear, you could say かのじょはわたしのむすこのむすめです。 The particle の shows the relationship between two people or objects. So you need to think about who or what you are trying to connect when writing. The second example: わたしのきょうはちちのたんじょうびです。 This sentence is also out of order. You need to put the word わたし and ちち together to show that he is your father. You do that by connecting the two of you with the particle の. Therefore, this is the correct order:　きょうはわたしのちちのたんじょうびです。 Also, keep in mind that if you introduce a time element with the particle は it usually goes at the beginning of the sentence. わたしのたんじょうびはあしたです。 This sentence is fine. It reads very clearly.

×　わたしのきょうはちちのたんじょうびです。　→　○　きょうはわたしのちちのたんじょうびです。

And is this right? Today is my father's birthday. My birthday is tomorrow. わたしのきょうはちちのたんじょうびです。わたしのたんじょうびはあしたです。