happily属于方式副词
在方式副词中，除了well,fast,hard等少数副词外，大都是以－ly结尾的，
在句中多取后位！
如：He did the work well!
He studied hard for the people.
方式副词在句中也可取中位：
如：He was happily playing with toys.
She slowly opened the door and locked inside.
在被动结构中，方式副词常取中位：
如：When I saw her, she was being well looked after.
Do you think the repairs have been properly done?
在个别情况下，为了强调，或其它需要，方式副词也可取前位:
如： Gently the doctor examined the child's egg.（强调轻轻地）
综上，一般来说你的句子应该用They are happily living toger now.是完全正确的。
用They are living happily together now.或They are living together happily now.在口语中也是可以接受的，但不是最佳。至于Happily they are living together now.在某些情况下，修辞的需要也可成立。
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