Those who don't understand why Emperor had to kill Wang Yoo, should remember that they live in time where "words holds power" especially for emperor's women - a little word can put them in dangerous position and kill them.Remember how Lady Pak got in trouble that started with "rumors"?SN went against Bayan thinking " Everything I'm doing is for you" TH killed WY thinking "Everything I'm doing is for you" WY is definitely the best man in this series, he is honorable, a hero, smart, kind,brave, skillful almost perfect.TH is the complete opposite but hey he did everything he can to protect SN while inside the palace and as what the characters said, the war in palace is far more scarier than the war in battlefield.
Imagine having to live your whole life in that kind of environment where no one was there to really support him and show him love, that's really depressing kind of life. He is stupid, weak,whiny,obsessive, possessive, coward and all. In the end, the only thing he could offer her is his empire and the tittle but in the end he lost it, but still SN loves him and willing to stay with him.
She grew up pretending and doing as what she is told.
If you will notice, whenever she is with Wang Yoo she is always on guard unlike whenever she is with Ta hwan where she can act like a child, she smile and play around with him and treat him however she like (another proof of when she choose to ride with him because she didn't want to bother the king but was fine doing it to the emperor which means she's comfortable around him) And again, she grew up having to follow what she is told, acting like a servant of the king despite his confession to her.
The king was so excited because she's asking a favor which is definitely rare or almost never happened, only for him to be a bit disappointed because that favor was actually for the emperor, because he doesn't have much appetite and that his clothes were ripped but doesn't have a sewer to fix it.
These are just small problems that probably even the emperor himself doesn't care about but she noticed them. Yom was the one who tortured her father and even if Ta Hwan didn't betray her, her father will still die and yet, the one she hated the most was Ta Hwan. because being betrayed by someone whom you trusted and cared for, will be more painful than those whom you didn't care about.
Jealous enough that she started thinking that she need power to become an empress and even asked WY for help.