There are many electron-positron colliders. LEP was the largest one so far. You can do particle physics with it, in a much cleaner way than with proton-proton colliders as the colliding particles are elementary. The downside is the lower energy of the collisions.
There are no neutron colliders as you cannot accelerate neutrons in an accelerator. They can make secondary collisions, of course, if they are produced by other collisions.
Sure, but it does not give the interesting results electron-positron gives. All the cross-sections are much lower, the reactions need much higher energy and so on. You also need two separate beam pipes (or a linear accelerator).
Fixed-target electron experiments are some low-energetic version for electron-electron collision, together with some electron-nucleus collisions.