It's a pre-snap read. Drew reads the defense, whether they're playing off the LOS and away from the sidelines, and checks to it. Bush doesn't get nailed "MOST OF THE TIME". Most of the time he gets 3-5 yards and goes out of bounds.
A swing pass is a swing pass. It's a route the running back runs. There are no "versions" of it.
No. That's not what it intends to do. The purpose of it is to force the linebackers wide or the CBs short. It's "stretching the defense". "Stretching a defense" doesn't mean sending everyone 30 yards deep. Then the defense just plays 30 yards off the LOS. Stretching a defense involves someone on the LOS, someone 30 yards deep, someone in the middle of the field, someone on the sideline. In otherwords, force them to defend the entire field.
Again, the swing pass is, more often then not, almost certainly a pre-snap checkdown by Brees at the LOS because of what the defense is showing.