The rear gate to the puddle jumper is blown open to allow the escaping air to push it through the stargate, yet when the jumper appears in the Atlantis gate room, the door is not down, and would not have been able to retract while passing through the wormhole. However, in a later episode when the hatch is blown, it completely detaches from the jumper into space, so there would be no door to be shown.
It's been well established that it requires a minimum of 7 "Chevrons" to dial any Stargate address, yet in the first scene when they dial the gate from the jumper, only 5 Chevrons are entered to open the gate.
The premise of the main conflict in this episode is that the gate only transports whole objects. Previously established gate physics would indicate otherwise. Examples include the MALPs arm being used to obtain intelligence through an open wormhole and Colonel O'Neill "holding the door open" by keeping his hand on the off world side of the gate when the Asgard came to reclaim their technology from the rogue NSA operation.
As the medical team are reviving Sheppard, you can see his abdomen rise, indicating that he is breathing on his own.