Charlizle didn’t have magic in her first life, the mana that made her powerful now was the mana that each emperor pushed into her as a sword. She must’ve been overpowered by multiple mages (those who conducted experiment) plus the threat that her family and fiance will die.
About the tutoring, Dylan is the 13th prince. He is faraway from the throne that is why they allowed even incompetent people to tutor him (this is maybe to mock Dylan too) since they don’t really care abt him. The other prince felt threaten because when Dylan accepted Charlizle as his tutor, he basically announced that he has interest in succeeding the throne. That is how they perceived his action. And since he is too faraway the thrown, the emperor doesn’t care if he died, since he has a lot of sons.
The translated novel isn’t too faraway in the story yet (at 45 chs). But from what I understood so far is that, Charlizle doesn’t want to rule the empire herself. She just want to get her revenge and scram. So she chose Dylan to become emperor. She is the only one who knows how gifted Dylan was and he was overshadowed by the 5th prince. Basically, 5th prince became emperor but Dylan was the brain behind his ascension. Maybe she sympathize because like her, Dylan’s skills was never recognized.