Neither Jack nor Joe have oxygen bottles on their backs when they enter the elevator with the rescue team, but the have then when they get off.
When Henry is doing the autopsy on the heretofore unknown body, he states that "she has been exposed to an intense dose of radiation" implying that he knows that the body is female, and what happened.
In the very next scene, Carter says that they can rebuild their John Doe on a cellular level implying that they don't yet know that the body is female. That is because Henry knows who the body is and no one else does.
Jack goes back in time to fix the time line. He has to prevent Henry from doing what he does to change the time line. Henry knew back in the old time line that what he was about to do would change the time line and ultimately save the woman he loved because Henry had gone back in time first.
Right after the tornado destroys the building, Joe says "it's just the beginning" as if she knows what is to come which is impossible because she knows nothing about the cause of the anomalies.
There is no logical reason for the control panel to be in the same room as the artifact instead of behind the shield where Stark is standing.
Nathan says the artifacts technology is light years beyond our own, but as a Physicist he would know that the light year is a measure of distance, not time. The light year is the distance light travels in one year, about six trillion miles at 186,000 miles per second.