Typically Jeannie is shown crossing her arms and nodding to perform her magic. Is this a needed action? Or are there examples where Jeannie performs magic without the gesture?
It's already confirmed that Jeannie can do her magic without crossing her arms, but apparently she does need to blink. From the season five episode "The Blood of a Jeannie":
As genies have red and green blood, Roger masquerades his arm as Jeannie's so that she can get a certificate of perfect health to be eligible to marry Tony. A man at the jeweler's store steals Tony's wallet so Jeannie blinks the jeweler's store into his office. Roger stops anyone leaving as they will walk out into a NASA corridor. Jeannie's eyes have been dilated and she can't blink but when she can she makes the thief return the wallet and blinks the store back. Roger apologizes and leaves when a man walks in from the street.
No. As you can see in this video clip, Jeannie can perform magic without crossing her arms.