I don't know any exact grammatical reason, but to me it sounds awkward when phrased that way (though I'm not a native speaker of German). I know that you can say 'Gemeinsam haben sie einen Sohn und eine Tochter', though.
According to https://en.wiktionary.org/wiki/Sie, Sie may also be used as the plural "you". I also gave the answer, "Together, you have a son and a daughter," but it was incorrect. I think it should be correct, though.
No, because people can have children from previous marriages or relationships. Saying "They have a son and daughter together" is better English because it implies that both the children come from the same parents.