What you are describing is similar to the electrolytic refining of crude copper to make high purity copper. This isn't "electrowinning" - that process is the electrolytic production of a metal from its ore, which is quite different and more complex.
The electrolytic refining principle is simple - two copper electrodes in a copper sulfate solution, with a direct current run between the electrodes. Copper metal is oxidized at the anode to Cu2+ ions, and these are reduced back to copper metal at the cathode. The copper sulfate solution is not consumed - it's concentration does not change - it just acts as a medium for copper ion transfer.
A block of crude copper (containing various impurities) is used as the anode. A thin strip of pure copper is used as the cathode. Since only the copper dissolves and is transferred to the cathode, this builds up pure copper at the cathode. Impurities drop off the anode as they are exposed and fall to the bottom of the cell.
You are doing more or less the same thing with your silver. Of course you have to use silver nitrate as your electrolyte, because that is one of the few soluble silver salts. The only other alternative is silver perchlorate, which you really don't want to be playing with.
So the process you are describing does make perfect sense, and would indeed work. Putting a filter around your anode is a very good idea, because if there is still any copper impurities present (unlikely, but possible), then these could react with the silver nitrate solution, displacing silver and contaminating the solution with Cu2+ ions. Some of which could be reduced at the silver cathode, contaminating it with copper.
From the process you described for making your silver anode, it's unlikely that there will be much (if any) copper contamination. So when you try this, see if you do collect any insoluble particles on the filter. If you don't, then the filter probably isn't necessary.
As to your questions:
(1). How much power to use? The voltage you apply only needs to be enough to overcome the reduction potential of silver, which isn't very much. 3 volts would certainly be enough, but a voltage that low will not drive much current through the solution.
Current is what matters for the rate of electrolysis. The higher the current flow, the higher the ion flow, so the greater the rate of silver transfer from the anode to the cathode. I have done electrolysis of a similar nature before (with copper or zinc), and to get a reasonable current flow, the voltage needs to be more than 3 volts. I used a power supply of my own design, which supplied either 12 or 24 volts. I found that with 12 volts, I could get about 5 amps to flow, and with 24 volts, about 10 amps. Of course, the higher the current, the greater the heat produced - at 10 amps, a couple of litres of solution will start to boil in less than 2 hours.
(2). Concentration of the electrolyte is also a limiting factor. To get the maximum possible current flow, you want the electrolyte to be as concentrated as possible - a saturated solution. Silver nitrate is highly soluble in water - 2500 grams will dissolve in 1 litre of water at room temperature. So making a saturated solution could be quite expensive.
However, the solution doesn't need to be anywhere near saturated for the electrolysis to work. A 1.0 Molar solution (170 grams per litre) is more than enough to get some current flow, and the process will still work (albeit at reduced rate) at concentrations far lower than that.