During the surgery, a women calls out "... he's breathing on his own" and they start cheering and clapping. This is not accurate. With the chest wide open, it's clear the operation is still going, hence he would be under a general anaesthetic and intubated with a machine doing the ventilation because you cannot breath on your own in these conditions. They wouldn't bring a patient out of anaesthesia until after they close them up.
During the day scene at the end of the episode where Budd and DiNozzo are on the phone they suppose to be in Shanghai. However the moment DiNozzo starts chasing Budd you can clearly see an American orange bus in the background.
It is said that the North Korean sub is "1200 ft below". Submarines must rise to a depth of less than 150 ft, so the arming of the missile wouldn't be considered an immediate threat until the sub started to surface.