Effect of a change in quality on price elasticity of demand
It seems that your confusion is more about classifying goods into necessities vs. luxuries. In economics, the criterion for such classification is to look at whether the income (not price) elasticity of demand is "high" or not. If a good has a high (and positive) income elasticity, then it's a luxury; if it has a low (but still positive) income elasticity, it's a necessity. What counts as "high" or "low" is an empirical matter. Investopedia suggests $1$ as a threshold (high if $&gt;1$ and low if $&lt;1$). 



Coming back to your question about price elasticity and quality improvement. The answer is: it depends on how you model the demand curve of the good. Recall that the price elasticity of demand is given by 
\begin{equation}
-\frac{\mathrm dQ}{\mathrm dP}\frac{P}{Q}=-\frac{1}{\text{slope}}\frac{P}{Q}
\end{equation}
where the slope is evaluated at the point $(Q,P)$. The quality improvement shifts demand curve to the right (since consumer is getting more value for each dollar spent, so they are willing to pay more for each unit or buy more units at the same price). If price remains unchanged (as you assumed), then quantity demanded must increase, say from the original $Q$ to a higher $Q'$. 

If the slope of demand stays the same at $(Q',P)$, then price elasticity would fall according to the formula. If demand becomes much flatter at $(Q',P)$ (with a magnitude greater than the increase in quantity), then price elasticity would increase instead.