Though the movie heavily implies that the prince had killed a woman before, but they were vague enough to make you really wonder about it. He did like to beat women up for sexual reasons.
Although that's as far as words could go in those times: Nobody would dare admit publicly that anyone of royal blood would have killed someone, especially a woman.
And the couple certainly relied a lot on coincidence in order to achieve their goals.
Well, he was staging a coup — those rarely end bloodlessly, and it may have caused the Prince to become a bit of The Caligula.
On the other the illusionist is basically guilty of murder because he wanted to be with the love of his life.
Why does the police inspector smile when he realises that he saw a man commit suicide in front of his eyes and had his own life threatened for a lie?
First, the crown prince was still a tremendous jerkass who beat women and there's a good chance he had killed a woman before, so he probably deserved his fate. Second, even if you ignore the whole woman-beating, woman-killing thing (since we don't know for sure that he's killed a woman), he still conspired to overthrow the Emperor. I'm sure the Emperor would be more concerned about that than his love life. Even if you took all the women-beating and killing stuff out of the equation, I'm sure the Emperor would see it fit to see to it his son is punished for conspiracy. That's why he sent the guards to arrest the prince, and that's why the prince committed suicide, so he certainly deserved all that. And third, I don't think it was a happy smile so much as it was a "that son of a bitch, I can't believe I fell for it, I can't believe he pulled it off, all I can do is laugh at this", smile.
He witnessed the greatest illusion ever done, the last and greatest trick by the finest magician in the world. And all Uhl can do is applaud the brazen courage and talent it took to pull it off, right under the nose of Uhl and an entire nation.
Exactly this. He's the witness — the only witness — of the greatest trick ever pulled. And on top of that, the master illusionist respected him enough to give him the blueprints... in essence, to say "you're a worthy enough mind to follow in my steps."