Do neutrons attract each other?

They interact using the nuclear strong force, and gravity.

 

With the strong force I don't think they attract each other at any range, but I can't recall, someone else will have to comment on this.

 

The gravitational force between any 2 energetic (or massive as the two are related) particles is always attractive, but this is very weak.

If protons and neutrons can gather in the center. Is it possible that the same effect could happen with electrons and neutrons, with protons orbiting the electron nucleus?

I believe the strong nuclear force, which is responsible for the nucleon bonding does not act strongly enough on electrons to break the electron coulomb repulsion (EM).

If protons and neutrons can gather in the center. Is it possible that the same effect could happen with electrons and neutrons, with protons orbiting the electron nucleus?

 

No. You can't confine electrons a such a small volume, and protons are much more massive — if we simplify it to a classical picture for a moment, Newton's third law is going to dictate that the more massive particles be closer to the center of the "orbit."

could the electrical pulls supplied by the electrons even hold the protons in orbit?

could the electrical pulls supplied by the electrons even hold the protons in orbit?

 

Ignoring the center-of-mass and quantum issues, yes. The force a proton exerts on an electron is identical to the force an electron exerts on a proton. It has to be, by Newton's third law. It would be true even if they did not have the same magnitude of charge.

 

You get a little closer to examining this when you create "exotic atoms" like muonium, the bound state of an antimuon and electron. The muon mass is smaller than that of a proton, so the orbital dynamics are changed, making it a good test of QED. Or a mesonic atom, where you replace an electron with a meson. Or positronium, the bound state of an electron and positron. And others.