Will the small magnet slide across the big magnet? 
If there are walls on the sides of the long magnet to stop the small one from falling off the short edge, will the small magnet launch itself across the big one?
Red is north, blue is south.
 A: Unlike the electric field that is created by monopoles (individual positive and negative charges), the magnetic field is created by dipoles - atoms with a magnetic momentum. Each atom is like a small magnet with two poles (N and S) oriented vertically on your pictures. You can think of a magnet as consisting of many small magnets joined together.
Imagine that the magnet in your picture consists of 100 thin magnets counting from bottom to top. Their combined field creates the poles of your magnet where N and S have the same strength, as on your left picture. When you cut your magnet in half by thickness, as on your second picture, you will have only 50 thin magnets left at any point counting from bottom to top. Their combined strength will be twice smaller, but N and S would still have the equal strength.
When you cut the material "off the N pole", you do not remove the N monopoles (like positive electric  charges). There are no magnetic monopoles - they do not exist. You simply remove a half of the dipoles (atoms). So you don't remove just N, but equally N and S of each dipole (atom).The result will look like this:

When you remove the cut off parts, you new setup will look again like from what you started, only twice thinner. Magnets certainly attract and repulse and can generate physical movement, but this specific design will not work. Sorry :)
A: If you cut the magnent diagonally, it will still have a north and south pole, the field in your third from the left will be stronger repulsive on the left  (red onred)than on the right so at most your cube will rotate, bringing south to north and sticking down. There is no momentum towards the left as you show in your image, momentum is a conserved quantity. The angular momentum (also a conserved quantity) for the rotation will be taken from the lattice demagnitizing the magnet accordingly. 
