The TV series "Women under the World" is adapted from the novel "Women under the World" by South Korean writer Park Jong-ho, so this story was invented for the sake of drama conflict. There is no doubt that Zheng Lanzhen in history is indeed Zheng Yunqian's daughter. According to historical records, her mother was born into an official family. As for whether her father was a big official before, as she said, there is no way to prove it.
Regarding Balingjun, there was no mention of Balingjun's name, nor did he mention whether he was a direct uncle of Zhongzong. Historically, the direct nephew of Zhongzong (the same clan as Dezong) was the monarch of Yueshan, so Balingjun was probably fabricated.