Will a bullet penetrate farther through 1,000 sheets of paper with no space between the sheets, or with space between them? Consider a gun pointed at (normal to) a 1,000-page stack of paper. Which would a bullet penetrate farther into in terms of number of pages: the stack as-is, or the same 1,000 pages stacked with spaces of, say, a millimeter of air fitting between them? Ignore gravity.
In the first case, it seems like the pages can reinforce each other, giving a stronger system. However, in the second case, the bullet must travel through both the pages and air, so through a longer distance.
Does the answer change if friction/air resistance is ignored?
Does the answer change if the pages are rigid instead of flappy?
 A: Ignoring air resistance, gravity, and a bunch of other variables here's an answer that will hopefully suffice:
Note: I use surface area interchangeably with volume as sheets of paper are very thin.
Why is a single sheet of paper easy to go through for a bullet? Well this is due to that large difference between the ability for the sheet of paper to resist the force of the bullet and the force of friction between the fibers of the sheet. now if the sheets are separated (say 10 cm between each sheet) the bullet will act as such: it will come in contact with the sheet with an area of lets say 2 mm the fibers of the sheet will resist the acceleration due to the force of the bullet. This will cause stress between the fibers which may interact in an area of slightly above the 2mm contact point of the bullet. The sheet will then rip when the force of the bullet overcomes the frictional force between the fibers or the fibers themselves tear.
However as we place the sheets close and closer together not only does the bullet interact with a single sheet but it simultaneously interacts with multiple sheets. This interaction of the sheets allows the force from the bullet to be more evenly distributed between the sheets in all directions. so the bullet will now come in contact with the first sheet which will resist the force of the bullet and send some of the force to the sheet behind it which then repeats again. As the sheets interact with each other not only is the force pointing straightforward but it is also being distributed vertically and horizontally between more fibers of the sheets allowing for more frictional force to resist the bullets force.  Since more area of the sheets are counteracting the force of the bullet it will be much harder for a bullet to go through a stack of sheets as opposed to when they are separated.
This is the exact phenomenon used in kevlar bulletproof vests. The fibers are very strong and the friction between the fibers is extremely great which allows for them to stop bullets by distributing the force to almost the entire vest.

Does the answer change if the pages are rigid instead of flappy?

If all the sheets are rigidly standing by being clipped to a string on all their edges the bullet will go through them more easily compared to if they are "flappy". The sheer stress and tension between the fibers of the sheet will be much higher and therefore they will be weaker as opposed to them not being rigid.

Does the answer change if friction/air resistance is ignored?

Now if air resistance is not ignored then we must state the distance between the sheets. I will assume the 1mm distance you proposed is the optimal distance (The sheets do not touch and interact but there is not a great amount of distance between the sheets allowing for air to become a major factor. At this point I would still absolutely say the stack of sheets is still much much stronger in its ability to stop the bullet for the reasons stated above.
I hope this answers your question.
An easier way to think about it is as if the bullet is an arrow. If it hits a single sheet it stays intact and the surface area of the sheet that interacts with the arrow is relatively small. However for a stack of sheets the arrow will have to distribute it's force through a larger surface area of the sheets  allows for the sheet the resist the arrow's acceleration.
