Bring water to boil in medium, heavy saucepan over medium-high heat. Stir in rice and salt. Return water to boil. Reduce heat to low. Cook, covered, until water is absorbed and rice is tender, about 20 minutes. Meanwhile, in medium pot over medium-high heat bring milk, sweetened and condensed milk, dulce de leche and cinnamon sticks to boil; cover and set aside.
Stir warm milk mixture into pot with cooked rice. Increase heat to medium-high; bring milk to boil. Reduce heat to low. Cook, uncovered, stirring occasionally with wooden spoon and scraping bottom of pot to prevent sticking and scorching, until rice mixture is creamy and thickened, 8-10 minutes. (Note that pudding will continue to thicken once cooled).
Pour rice pudding into serving bowl; serve warm or chilled. Garnish with pecans, pomegranate seeds or mint, if desired.