The water into the RO should be softened, not absolutely necessary but it will make the RO last longer and the results would be better. To use RO water for the entire house might be a bit wasteful as the RO gives out so much waste.
The DI is a final polish, it doesn't have to be in the van. I have a static system at home with WS RO DI and then pump it to the van which just has a tank. The water out of the brushes is always 0000.