Is this substitution correct? 
I think the second equation above is incorrect. It seems to me that the constant k will cancel out. Am I interpreting the equation for the price elasticity of demand incorrectly as equivalent to: 
$E = \frac{\partial q}{\partial p} * \frac{p}{q} $

Warning: I won't make the $t$ subscripts explicit.
The source you quote seems to have a misprint. I'm no economist, but$$E=\frac{p}{q}\frac{\partial q}{\partial p}\implies\frac{\partial\ln y}{\partial p}=\frac{1}{q}\frac{\partial q}{\partial p}=\frac{E}{p}.$$For constant $E$, this integrates to $\ln y=E\ln p+\text{constant}$. I suspect the author was thinking ahead to that step, resulting in $E\ln p$ instead of $\frac{E}{p}$. This is consistent with the first display-line equation after your excerpt, $$\ln y=\beta_0+\beta_1s+\beta_2\ln p+\varepsilon.$$This gives $E=\beta_2,\,\text{constant}=\beta_0+\beta_1s+\varepsilon$ (the integration "constant" need only not depend on $p$, but in this example depends, as it may, on $s,\,\varepsilon$).
