Byproducts of the neutralisation of hydrochloric acid with sodium hydroxide
Given a 1M concentration of $\ce{HCL}$ and $\ce{NaOH}$ here are the pHs:
  
  pH of $\ce{HCL}$: 1
  
  pH of $\ce{NaOH}$: 13-14


First, a 1 molar concentration of hydrochloric acid solution, $\ce{HCl}$, has a pH of 0.0 if we take ${-log[H_3O^+]=pH}$ to be true. (And it is mostly true for dilute solutions). 


  and of course water has a pH of 7.


Pure water at 25 degrees Celsius has a pH of 7.0. Boil the water and the water will no longer have a pH of 7.0. Cool the water and pH will similarly change. The only condition for neutrality is that the concentration of hydronium ion equals the concentration of hydroxide ion. It just so happens that water at normal lab conditions has a hydronium ion and hydroxide ion concentration of ${1.0*10^{-7}}$ M, which corresponds to a pH of 7.0. Your statement only holds at a specific temperature. At other temperatures, this reaction - the auto-ionization of water - which controls the pH of pure water - will go to varying extents:

$\ce{2H_2O &lt;=&gt; H_3O^+ + HO^-}$


  Here is a reaction that is most likely to occur:
  
  $\ce{HCl + NaOH + H2O &lt;=&gt; H2O + NaCl(aq)}$


Since we are talking about solutions of HCl and NaOH, your equation isn't reflective of what's actually going in on the system. HCl and NaOH both dissolve in solution (unless the solution is saturated with either). So you don't have HCl or NaOH in solution, but rather hydronium ion and chloride anions in the case of HCl and sodium ions and hydroxide ions in the case of NaOH solution. A more accurate equation would be:

$\ce{H_3O^+ + HO^- -&gt;2H_2O}$

Regarding:


  $\ce{NaCl(aq)}$


Sodium chloride has a very high solubility in water and dissociates unless the solution is saturated. So it would be more accurate to write simply sodium ions and chloride ions, instead of "aqueous" sodium chloride (unless you are referring to the ion-pair effect). Ion pairs often occur in solution; the name describes exactly what is implied - two oppositely charged ions are attracted to each other. For sodium chloride, ion pairing is likely minimal, but in other cases, such as between a carbocation and a halide anion, the ion pair effect has noticeable effects on chemical reactions (i.e. partial racemization rather than full racemization). 


  Like for example theoretically sodium oxides could form from the
  $\ce{OH-}$ taking a proton from another hydroxide anion thus giving
  you $\ce{O^2-}$ which could react with the sodium to give you
  $\ce{Na2O}$ which is a sodium oxide. 


So hydroxide ion reacts as an acid and abstracts a proton from another hydroxide ion. 

This is at least theoretically possible, but thermodynamically unfavorable. The oxide anion is very unstable in water solution to the point of being able to react with the solvent (water) itself to a large extent.