The shear and tension stress on the material would be the same, because only the difference matters for that. So for testing strength of the structure the cases are equivalent.
The material would still be under the total pressure though, so if the material itself can't withstand that pressure, it would degrade (it may change crystalline structure, compounds may undergo chemical reaction and such with corresponding change in modulus of strength; or it may just break on inhomogeneities and crumble). However solid material generally withstand significantly higher pressures.
And under water there is also higher difference between pressure on the top and bottom surface and corresponding difference in buoyancy.
A few things happen. One, the paper would bend, but lets pretend it's rigid, what happens when you lift. Your assumption is correct, air moves very fast, reducing but not completely eliminating the difference in air pressure between above and below. Air (molecules) moves at about 1,000 miles per hour.
http://www.phy.mtu.edu/~suits/SpeedofSound.html
For typical air at room conditions, the average molecule is moving at
about 500 m/s (close to 1000 miles per hour). Note that the speed of
sound is largely determined by how fast the molecules move between
collisions, and not on how often they make collisions. This is because
no energy is lost during the collisions. The collisions do not "slow
things down"
So, in theory, if you could lift rigid paper at 1,000 MPH, you might approach the air pressure on that paper of 1,500 lbs, perhaps even exceed it due to compressive forces above (and there's probably people here who could do that math - I'd prefer not to try), but you'd only get that kind of pressure if you are able to lift faster, or about as fast as the air can move in below the paper.
Another thing to keep in mind about air is that it's compressible, so, even in a sealed tube where air couldn't get past what you were lifting, the air could compress so movement would be possible even if Air couldn't get past what you were lifting.
Best Answer
Yes, atmospheric pressure is primarily caused by the weight of the atmosphere above you. This is why the pressure decreases as your altitude increases.
A fluid (like air) under pressure exerts equal force in all directions while at rest. [Side note: Technically it exerts slightly more force downwards because of gravity. However, this effect is only noticeable on scales much larger than what we are dealing with here, so I ignore this effect in this answer.] So the air at ground level pushes equally on the ground beneath it and the air next to and above it. This is the pressure that you constantly feel. Now imagine yourself in a gazebo - a roof above you, but with open sides. Clearly, the air pushes on you from the sides. But the air above you is also pushing down on you with the same force per area. Why? The air above you is being pushed on the sides, which means it is under the same pressure that you are. It pushes back against the air compressing it from the sides and pushes in the vertical direction, both up, against the roof of the gazebo, and down, against you.
While rooms have walls, they are not airtight, which means that air can flow through them. If the pressure inside the room is different from the pressure outside, air will flow to "balance out" the pressure imbalance. In effect, any room is like a gazebo - the pressure will remain basically the same inside and out if nature/physics/entropy is allowed to run its course.
That being said, if you are willing to do some work, you can change the pressure inside a room - see
http://en.wikipedia.org/wiki/Negative_room_pressure
and
http://en.wikipedia.org/wiki/Positive_pressure