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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.
Hydrogen evolution will be favored by increased hydrogen concentration (lower pH). As the copper plates out, the Cu++ ion is replaced by H+. The very first plating out should be with the least hydrogen, and then hydrogen gas evolved should increase as the electrolysis continues. How can we reduce the pH without precipitating the copper?
I have not done the experiment, but I think that adding a bit of ammonia would convert the H+ to NH4+ and raise the pH enough to reduce hydrogen evolution without precipitating copper (Cu++ --> soluble Cu(NH3)4++). You might get more creative and use other amines.
Cell voltage is important also, as suggested in the comments. High current forces something to deposit; copper is preferred, but if the current demands, hydrogen will deposit also.
If you want to get really inventive, you might consider a cathodic poison that is not incorporated into the copper deposit. Mercury would work, but is a no-no for many reasons; sulfur and arsenic compounds also probably not good; maybe a long chain water-soluble polymer, or maybe even a sugar would coat the copper cathode and interfere with hydrogen deposition without becoming incorporated. Good luck!
Iron, Aluminum and Zinc are all significantly more reactive than Copper, so they will react with the Copper Sulfate and displace the Copper. I'm assuming that's not what you're trying to do, so you'll likely have to use an inert metal (Platinum, Gold, Silver) that conducts electricity well as your anode.
Edit: When I say that Iron, Aluminium and Zinc are more reactive, what I mean is they have a greater electropositivity compared to Copper and are, therefore, more likely to donate/lose electrons to form positive ions.
The first step should be to ask a question so that you can get an answer which gives you data that you can analyze and discuss. Research generally tries to ask a question that results in some improvement for the world. How to better electrowin copper would be a good question. It seems that you are not just looking to move copper from one electrode to the other, but to understand what might be going on within the solution, because, after all, impure copper in one place is not that different from pure copper in another place.
What is the question that changing anode materials will answer? Different anode materials may dissolve (or not) and change the composition of the solution, which could be desirable, or it might be considered a contamination. Different anode materials will give different voltage vs current curves, because active metals will assist the electrolysis.
In a commercial environment, purifying copper should be routine until the solution becomes so loaded with active metal contaminants that voltage adjustments have to be made to continue plating purest copper, and now a decision has to be made about disposing of the solution. More noble contaminants, like silver and gold would precipitate.
So what could you do with a solution of copper sulfate and ferrous sulfate? Zinc has been mentioned as a frequent impurity, but in large operations, nickel is the soluble impurity of considerable value https://metalrecyclingmachines.com/metal-recycling/metal-recovery-systems/copper-electrolytic-refining-plant.html Lower pH keeps some metal hydroxides in solution rather than precipitating, but will increase the electricity needed to transport a given amount of copper. Nickel sulfate may not be available for you, so I suppose ferrous sulfate could be used.
Perhaps you could pick one anode material and vary the solution to mimic the end of life solution loaded with impurities (in your case, ferrous sulfate). In this case, an iron anode would seem best. Then with variable ratios of copper to iron (and perhaps at a couple of pHs, since this could be changed to assist the process), what voltages, or probably better, what current densities (amps/cm^2) are optimal for squeezing the last bit of copper out of the solution? So the anode could remain the same composition, but vary in area. (You will also have to try just dumping in a wad of steel wool - skip the electrolysis. This is an out-of-the-box shot to just see what will happen. No guarantees.) I suspect that too low a current takes too long but will give pure copper output (as seen on a stainless wire cathode), while a high current density will get the copper out of the solution faster, but it might be contaminated (how? air, iron, solution - I don't know).
Good luck. Be safe.