Wondering if a sewer vent obstruction could cause a sewer gas smell in all the drains when the upstaire toilet is flushed? Problem didn't exist until a super flush toilet was installed upstairs.

Is there a lower toilet below that? If so, how is that working now? You may hear some bubbling or gurgle noices as well if it's the vent.
It may be that the seal is off at the connection of the flange of the toilet you just installed. But then - you said you were smelling it at all the drains in the house?
Would that include the kitchen? (kitchen often has it's own vent)

Could the suction of the new super flush toilet pull the water out of the other drains in the house. If the vent pipe is obstructed, it would seem that more suction would be applied to the other drains in the house

Yes, an obstructed vent stack may not allow air to enter the drains as sewage flushes down the pipe. This would cause the traps to siphon and allow gas to enter the home. Your super flush toilet did not cause the problem, the blocked vent did.