When Regina and Jefferson are trying to make the hat work in the vault, Regina points to the hat then pulls her hand back as Jefferson puts the hat down. When the camera cuts to a different angle, Regina's hand is still pointing to the hat and she pulls it back again.
When Emma is walking up to Regina's doorstep, her hair is just past her shoulders, her red jacket is slightly unzipped with a black shirt underneath, and she is wearing jeans. When Regina opens the door and for the remainder of the scene, Emma's hair is much longer, she is wearing a different style red jacket that is zipped to her neck, and she's wearing a skirt. It is likely the shot of her walking to the front door was filmed for earlier in the season but was recycled and used for this scene.
When Regina uses Jefferson's hat to retrieve the apple, she is able to travel back to a specific moment in time to summon the apple from the past. Later seasons make it clear that nobody can / has traveled through time. However, Regina does not actually time travel, only the apple does.