Preheat oven to 325 degrees F. For rice pudding, in a large mixing bowl beat together the eggs, half-and-half, milk, the 1/2 cup sugar, and vanilla with a rotary beater or wire whisk. Stir in rice. Pour into a 1-1/2- or 2-quart oval baking dish. Place dish in a roasting pan set on a rack. Carefully pour 1 inch of boiling water into the baking pan. Bake, uncovered, for 45 to 50 minutes, until a knife inserted near center comes out clean.
Meanwhile, for the chocolate sauce, in a small saucepan stir together the 1 cup sugar, cocoa powder, chopped chocolate, and salt. Stir in water. Cook and stir over medium heat until chocolate is dissolved and the sauce comes to a simmer. Remove from heat. Stir in molasses and the 1/2 tsp. vanilla. Transfer sauce to a serving bowl; cover with plastic wrap.
As soon as rice pudding is removed from oven, stir and swirl in some of the chocolate sauce. Serve rice pudding warm with lingonberry sauce and sweetened whipped cream. Pass remaining chocolate sauce.