Surely you've the answers to these questions, Cinderella. You have spent as many as a few hours with the man while under the impression he was someone else. That ought be plenty of time to know such things.
For the sake of brevity, I didn't include another reason for Prince Elliot hiding his identity: He wanted Cinderella-Tedd to feel she could speak freely with him. Even if it wasn't for brevity, it could feel a bit odd to bring up now. Very few characters have shown Prince Elliot much deference.
Heck, it's likely canon that he helps out with cleaning the palace when he's available.
"YOUR MAJESTY! Are you helping the help again?"
"I've no idea what you mean."
"You're holding a feather duster behind your back!"
"I was... selfishly pretending it was a magic wand! For my own frivolous amusement!"
"LIAR!"