Henshaw's earpiece changes ears when he is in the room with Alex.
Kara says that she doesn't need to go to the hospital when she breaks her arm because when she gets her powers back it will heal. Her arm would still need to be put in a splint or cast, otherwise the bone in her arm would heal incorrectly causing potential disfigurement and later complications.
It is mentioned in a previous episode that only Kara's palm print will open the door to Alura's projection room, yet the DOE agents confer with her before seeking out Reem, seemingly without issue.
This is supposed to be the first time Kara has been cut or injured since she is not invulnerable right now, yet her ears are pierced so she had to have her ears subject to being pierced at some time on Earth since she didn't have them when she left Krypton.
No one questions why Kara no longer has a broken arm or is no longer sick even though the episode takes place in the events of 2 days.