Converting from AC to DC usually uses the standard conversion ratio: DC x 1.414, or AC/ 0.707. This is standard throoughout the industry, so your values look good if you are doing a single wave conversion. Standard losses for a silicon diode are 0.7 volts, and that doesn't depend on their size in amperage.
If you do add a capacitor, it goes across the DC side, not the AC. This will raise your apparent votlage, but not your current. It will, however, help filter out any noise.
I must add, your wave form looks very good, but the sharp edges will give you a bit of harmonics. The capacitor will help that. You will find that the apparent voltage will "sag" somewhat under load.
Since you are only at about 10 volts, a 25 volt capacitor at, say, 50mfd, would do the job, but you might want to add a smaller one, say at 0.1mfd, just to help the noise filtering. I would recommend all capacitors at a 100 volt level just to take care of any "surges". That's safer and they aren't that spendy. The different size capacitors do different jobs. One is low frequency, the second, higher.
Good luck, looks like you're going in the right direction.
Very good question.
