Side products during the electrolysis of Copper Sulfate?
There are a few key factors that I expect are contributing here:


$\ce{CuSO4}$ concentration – If your $\left[\ce{Cu^2+}\!\right]$ in solution is low, then that will disfavor deposition, as permeakra noted in a comment. You want at least $25$ to $100~\mathrm{g\over L}$ of copper in the electrolyte.
Cell voltage – $15~\mathrm V$ is an exorbitantly high voltage for copper electrodeposition. In order to avoid gassing at the cathode, you'll want to operate down in the $1.5$ to $2.0~\mathrm V$ range, tops.
Electrolyte conductivity – you don't mention anywhere that you've added other constituents to try to increase the conductivity.


If this is a home experiment, I would try dissolving the $\ce{CuSO4}$ in table or pickling vinegar instead of plain water. That will give you more charge carriers in solution, and also decrease the $\mathrm{pH}$, both of which should help.
If this is a school or work experiment with access to proper lab equipment and chemicals disposal services, I would use $1\%$ to $10\%~\ce{H2SO4}$ as the base electrolyte. That will really get your conductivity up, and your cell voltage down.



Finally, on the methods side, I would recommend choosing your voltage in order to initially target a current density in the vicinity of $\sim\!10~\mathrm{mA\over cm^2}$, as a value where I would expect to see little to no $\ce{H2}$ evolution. You can always nudge the voltage up from there as needed.


  Is $\ce{H2}$ generation happening as a result of the weak sulfuric acid in solution?


Almost certainly not.


  Can you calculate the products of simultaneous electrochemistry reactions?


Not with much certainty. Most of the time the best one can do is make qualitative arguments about what reactions might be happening.  Detailed modeling of all reactions occurring at an electrode is still prohibitively difficult, as far as I know.