You can use the same beets for the first and second batch, in the second batch you can use the some of the juice from the first batch rather than whey. The second batch will be slightly less strong. Throw out the beets after the second. You can reserve some of the liquid and use instead of the whey for future batch.
I would still strain your yoghurt to make sure you are separating out the milk solids.
As to detox, don't know, everyone is different. This is a good digestive aid and blood tonic, as well as, cleanses the liver. It's recommended 4 oz glass morning and night, you might want to start out with 1 or 2 oz and build up to see how you react.