When Rory is on the phone while she is at the hospital with Logan she has a flip phone. However, in the previous episode, Gilmore Girls: I Get a Sidekick Out of You, Rory's dad, Christopher, gives her a Sidekick (which is not a flip phone).
The scene where Rory talks to Logan after Logan's father comes to see him is not shot in two different places. You see many lamps and other miscellaneous items in the background while they are talking, but these are all gifts and items have been brought to Logan's hospital room. The background behind Logan's head is clearly the wall in the hospital room, which is why he still continues to say that they are in the hospital. The pull back shot at the end is also of them in the hospital room.