I don't think Regina/EQ knows everything - she must not know Emma's true identity as Snow & Charming's daughter. True, she's not very nice to Emma, but that's just because she doesn't want Emma taking Henry away from her. If Regina/EQ knew that Emma was Snow & Chaming's daughter, she'd be treating Emma a WHOLE lot worse.
What *we* now know is that both Regina/EQ and Gold/Rumpel know about the curse and who everybody really is in Storebrooke; and each one knows at least one thing that the other doesn't: Rumpel knows Emma's true identity (and Regina doesn't), and Regina knows that Belle is alive (and Rumpel doesn't).
Oh, Regina knew Gold was Rumpel all along. Perhaps she wasn't positive that he knew about the curse, and asking his true name was her way of finding out for sure. Plus, Rumpel seems to gain some "power" or something by knowing people's names, so by forcing Rumpel to tell her his name, Regina/EQ was trying to show she had some power over him.