Why doesn't hydrogen have a neutron? Why doesn't hydrogen have a neutron?
 A: Your question boils down to why didn't protons in the very early universe combine with neutrons to form deuterons (a proton plus a neutron).
The answer is that they do - there is a brief window of time where deuterons can form and the universe is cool enough that they don't immediately break apart. However, the system is in thermal equilibrium, thus seeking to minimise the total energy density; although the deuterons are stable, helium is a much more stable nucleus and as a result the deuterons rapidly combine to form helium and this mops up all the available neutrons in the early universe. Thus the vast majority of hydrogen is in the form of its neutronless isotope.
Protons can get subsequently converted into deuterons during nuclear fusion in stellar cores. The formation of deuterons is the first stage in the pp-chain of nuclear reactions that turn hydrogen into helium. As such, a significant fraction (maybe 10-20%) of the hydrogen in most stars does get transformed (briefly) into deuterium during their main-sequence lifetimes, but again the deuterons are then quite unstable to subsequent reactions that rapidly synthesise helium. The net effect of stellar nucleosynthesis is therefore to destroy deuterons - both those produced in stellar nucleosynthesis and those produced in the early universe.
Thus there is very little deuterium in the universe because its production by fusion requires a temperature high enough to then allow further fusion to produce the more energetically favourable helium. That would require some non-equilibrium process that produced deuterium-enriched material and then cooled it on short timescales. For similar reasons it is very difficult to produce deuterium by fission (e.g. spallation reactions). Here the issue is that disintegration to produce alpha particles and free neutrons is more energetically favoured. 
Similar arguments pertain to the tritium (one proton, two neutrons) form of hydrogen. That too can be formed in minute quantities in the big bang and inside stars or by spallation reactions, but here there is the additional problem that any tritium produced is unstable with a half-life of 12.3 years. Thus any tritium we observe in nature must have been produced very recently in energetic events (e.g. production in the atmosphere by cosmic ray collisions).
A: Neutrons usually act as buffers in nuclei. Protons are positive and repel positive things that are near them. Neutrons, with no charge, thus act as buffers, lessening the amount of repulsive force. The residual strong force keeps protons together too, but that’s not what this question is about.
Since neutrons act as buffers for protons (keep in mind the plural), there is no real need for them in Hydrogen, which only has one proton. There is no need for buffering. Hence, a lot of Hydrogen doesn’t have any neutrons, since it simply isn’t needed for Hydrogen to exist.
Now, Hydrogen can exist with neutrons, though the amount of Hydrogen with neutrons is dwarfed by that without. The most common isotope is Protium, with no neutrons. Then there is Deuterium, with one neutron, and then there is Tritium, with two.
A: For deuterium to form, first two protons must overcome their electrostatic repulsion to fuse to form a helium-2 nucleus, which will then undergo beta decay. Unsurprisingly, most hydrogen doesn't do that. Deuterium is mostly a Big Bang byproduct, which if anything is net destroyed in stars.
