The father and son likes music.

第一句对的,因为第一句用了一个定冠词,表示为这位既是作家又是老师的人.
而如果是The writer and the teather 则用are

第二句错的,第二句因为儿子和父亲不可为一个人,所以只能表示为复数形式:the father and son like music.

不对
复数形式
the writer and teacher are going to come.
the father and son like music.

第一句正确，因为The writer and teather 可以是同一个人的两种身份，第二句不对，因为The father and son 不能代表同一个人。likes应该改成原形，用like。

两句都有问题!!!
应改为：
1.The writer and teacher are going to come.
2.The father and his son both like music.

Both of the sentences are correct.The first sentence means that the person, who acts as a teacher as well as a writer, is going to come. The second one means that man,who is the father of his son as well as the son of his father, enjoys music. In each sentence, the subject refers to the same person. In students' practice, the first sentence is requently seen and the second one is rare.

仔细推敲都没错，第一句不说了，2种身份，第2个的话也有可能啊，比如你爸对你祖父来说不就是儿子吗？