Did Prince Leopold pull out the sword in a single attempt?

Prince Leopold took it as a challenge to reveal the mechanisms behind Eisenheim's tricks out of his self-esteem. So he invited Eisenheim to his private party to show everyone, like you wrote, that Eisenham was a mere charlatan and nothing more. His plan was to spoil his tricks in front of the public, and this can be aptly done in his private place.

At the end of the show, we saw that Leopold failed in uncovering any method of the tricks Eisenheim had shown in the party. Even Eisenheim made him request to himself through eye-contact to let the sword go with Leopold.

It was a grave insult to Leopold. He made him look foolish in front of the public, even with no help from any gadgetry. So he felt offended. It came from both his inability to pull the sword out himself and unraveling the trick.

For clarification, Leopold could not pull the sword out if Eisenheim did not let it go. If you look closely, Leopold made an eye contact with Eisenheim, to which he nodded, and only then could he pull the sword out.

Eisenheim also connects his sword act to King Arthur, saying that only one worthy of being the king would be able to lift the sword. Everybody assumes this means Prince Leopold (including the prince himself).

However, like Mistu4u said, on Leopold's first attempt, he cannot pull the sword out. This is made clear by his glance to Eisenheim, warning him not to do what he's doing. Only after Eisenheim nods does Leopold try again and succeed in lifting the sword.

(To answer your question about the staggering, he was mostly likely putting in more effort than needed to lift the sword because he didn't know if it would lift easily or not, so when the sword did lift easily, he lost his balance. It's like if you're pulling hard on something and suddenly it gives. You would stagger backwards or even fall from your own momentum.)

Now, in itself the trick might not have been overly offensive, but it's what Eisenheim was suggesting at that was where Leopold took offense. By saying that only someone worthy to be king could lift the sword, then blocking Leopold from lifting the sword on his first try, Eisenheim seems to be suggesting that Prince Leopold is not fit to be the emperor. He seems to also be suggesting that he himself is more powerful than the prince. This is why Leopold reacts the way he did. He bans Eisenheim from performing because he wants to destroy anybody that might get in his way of becoming the emperor.

Leopold was surprised that Eisenheim had the nerve to show him up.

Leopold was electrocuted slightly, you can hear the zapping sound of electricity as he pulls the sword off the ground, and the dismay in Leopold's face as he is electrocuted, perhaps a shock from the mechanisms holding the sword down. This clearly surprised Leopold and agitated him. As Mistu4u noted, Leopold gave Eisenheim an icy glance and he relented, allowing Leopold to withdraw the sword. Eisenheim knew Leopold had it out for him and he couldn't help but taunt him.