Okay, so the basic plot is that before the suitor comes, Turnadot had been executing various other suitors/princes because they have failed to answer her riddles three. In the beginning, we see her having a Persian prince beheaded. Here is what troubles me: Why isn't Persia getting all up in arms about her lopping off one of their son's heads? I know it's just a play/opera, and I need to relax, but it's just a bit of Fridge Logic when you think about it.
They might have accepted it as Fate, the prince knew what he was walking into, so if he failed he has only himself to blame. Seems to happen a lot in folk tales.
More pragmatically, China and Persia are, to each other, foreign, exotic distant lands (there's a reason so much of the 1001 Nights and Sindbad are set in or from China), chances are they wouldn't know for a long while if ever beyond that a prince died in foreign distant lands.
The story comes from a 12th-century poem very much like 1001 Nights, so this makes contextual sense.
Not to sound sexist or anything, but the supposed "Grand Emperor" really has let his daughter walk all over him. If he didn't like her idea of 'killing off suitors', why didn't he tell her something like, "You will marry this man! I do not care if you like him or not, you will marry him and you will give me an heir!" Again, I don't mean to sound sexist, it's just that...his behavior is kind of odd for a grand emperor...
Maybe he thinks Turandot has the right idea? He figures the one who finally passes her test will be the only worthy one.
In his brief statement before the riddles, the Emperor says he has taken a terrible oath that binds him to support Turandot's cruel demands. Possibly Turandot's bloodthirsty rampage is legally valid under the auspices of the oath even if the Emperor swore to it for something else.