The Denver Airport has a pretty distinct roof to represent the Rockies. When the Duncans are leaving Denver, the airport shown is Reagan Airport in DC - the Washington Monument is visible in the background.
Teddy is filming Charlie putting the star on the tree, as seen through the camera's eye. Teddy then puts the camera away. Charlie is then seen through the camera's eye again.
Teddy said the flight to Palm Springs would be Charlie's first ride on an airplane; however, Charlie, Teddy and Amy flew to Chicago in "Charlie Shakes It Up" and during "Sun Show, Part One", the whole family took a plane to Hawaii.