All the data I've seen to date support this strategy. I think there are certain groups in the population that need to be considered, especially the elderly and those with immune deficiencies. They may need to have a second dose sooner, or as soon as possible.
Regarding the other question about the immunity obtained after infection, this is a natural infection. There was a study done at the University of Montreal by Professor Andrés Finzi, who followed up on a population of patients who had been infected. I think they have more details now, but a few weeks ago it was concluded with certainty that for the first eight months there was strong immunity related to long-term immunological memory. That doesn't mean it's complete immunity, but it's very strong.
The study hasn't yet determined the rate of infection among these people, but from the immune tests, we have important data showing that the immune system is still very effective against new infections.