When does the 'iddah (waiting period) for a woman whose husband died by drowning begin if she learned of his death after some time, and is she obligated to reside in her in-laws' home for the sake of her son?

The waiting period (iddah) for a widow is calculated from the day of death, not from the time the news reaches her. The iddah must be spent in the marital home, unless doing so would necessitate mixing with men. There is no obligation to reside with the husband's relatives after the iddah ends.