Does light entering a multimode fiber undergo a polarization change during propagation through the fiber? If so, can the emerging light be linearly polarized by placing a polarizer at the fiber’s output end?

If you introduce linearly polarized light into a multimode fiber it will become randomly polarized at the exit face because of the many internal reflections it undergoes within the fiber; the same is also true for singlemode fibers. Adding a polarizer to the fiber’s output end will turn the randomly polarized light into the linear that you desire. Polarization maintaining fibers exist as well, though Edmund Optics® does not offer any.