Why does this quiz question say that protons and electrons do not combine to form neutrons? I read this somewhere:

Where are the protons and electrons in a neutron star? When the neutron star forms, most of the protons and electrons combine together to form neutrons. 

But on a true/false quiz, I saw the question

A neutron is formed by an electron and a proton combining together and therefore it is neutral. 

but the answer was false. Why are these not contradictory?
 A: You're asking about two distinct phenomena.  The difference between them is subtle, and I think there is some context missing from the second question that you quote, which makes things more confusing than they need to be.

When the neutron star forms, most of the protons and electrons combine together to form neutrons

This is mostly correct.  The process is known as "electron capture," and the full reaction is
$$\rm p + e^- \to n + \nu_e$$
The other particle in the final state (represented by a nu) is a neutrino.  The neutrino is an uncharged, very low-mass electron-like particle, in the same way that neutrons and protons are different charge states of the same sort of particle.  So far as we know, in physics, the number of electron-like "leptons" and the number of proton-like "baryons" isn't changed in any physical process.  The neutrinos play an important role in the dynamics of the stellar collapses where neutron stars are formed, but in some authors who write very elementary explanations of neutron stars will leave the neutrinos out of their descriptions.  There are advantages and disadvantages to this approach; your confusion here is one of the disadvantages.

A neutron is formed by an electron and a proton combining together, therefore it is neutral: true or false? Answer: false

This is a fundamentally flawed true-false question, because it makes several statements at the same time, some of which are correct.  The question I was expecting to find here, based on the title of your question, was more like

The neutron is an electron and a proton that are "stuck together" somehow.  (Answer: false)

We have another name for an electron and a proton that are semi-permanently "stuck together," and the dynamics of that system are very different from the dynamics of the neutron.
When you "combine together" macroscopic objects in ordinary life, the things that you combined are still somehow present in the combination.  But in particle physics, the situation is different.  The electron-capture process that we're talking about here fundamentally changes both the baryon and the lepton parts of the system.  To the extent that a neutron behaves like a composite particle, it behaves as if it is made out of quarks.
A: It's not exactly a myth that protons and electrons combine to form neutrons, but it's not very accurate. A proton and electron can react to produce a neutron, but a neutron isn't simply a composite particle consisting of a proton joined to an electron. 
Protons and neutrons are hadrons, which means they consist of quarks. A proton has 2 up quarks & 1 down quark, a neutron has 1 up quark and 2 down quarks. Quarks are bound together by gluons (and a bunch of virtual quarks, but don't worry about them for now). Hadrons consisting of 3 (or a higher odd number) quarks are also known as baryons.
When a proton & electron react an up quark in the proton is converted to a down quark, and the electron is converted to a neutrino. This process is mediated by a $W^+$ boson. I can't find a good diagram of this exact process, but here's a diagram from Wikipedia of a closely related process: the decay of a neutron into a proton, electron, and antineutrino.

The diagram for the proton + electron reaction is very similar, just reverse the time direction, and swap the $W^-$ to a $W^+$, and the antineutrino to a neutrino.
