There are three dogs Bello, Max and Rex, according to these sentences, Bello is masculine and Max is Feminine, but what about gender of the whelp Rex?
Yes, there are three dogs. However, only the names of the dogs and the use of the masculine form of 'Hund' imply their gender - not the possessive pronouns. Furthermore, it is not evident from the text that Rex is a whelp.
I'll explain the separate sentences in detail below:
Bello ist der Hund ihres Mannes.
Bello is a dog, belonging to a man that is the husband of a woman that is referenced
Rex ist der Hund seines Kindes.
Rex is also a dog, belonging to the child of a man that is referenced
Max ist der Hund ihrer Mutter.
Max is a dog, belonging to the mother of a woman being referenced
Rex ist der Hund seiner Kinder.
Rex is a dog, which belongs to the kids of a man being referenced