Preheat the oven to 350°F. Line a large, rimmed baking sheet with aluminum foil and place the ribs on it. Add salt and pepper, to taste, and dust with the garlic and onion powders. Cover tightly with aluminum foil and bake for 1 hour.
Line a second rimmed baking sheet or roasting pan with aluminum foil. Set aside.
Pour the root beer and coffee into a large saucepan, bring to a boil, and boil rapidly until the liquid is reduced by two-thirds, or measures about 11⁄4 cups. (To check, pour the hot liquid into a 2-cup glass measure.) Add the ketchup, 1 tablespoon of tomato paste, lemon juice, and mustard and whisk to blend. Add salt and pepper, to taste, and bring to a simmer. Cook for 5 minutes, stirring occasionally. Remove from the heat. The sauce will thicken as it cools, but to make it thicker or give it more tomato taste, add the remaining 1⁄2 tablespoon of tomato paste and whisk to blend.
Remove the ribs from the oven (expect a lot of fat in the pan) and transfer them to the other foil-lined pan. Spoon some glaze over each rib. Reduce the oven to 325°F and bake, uncovered, for 50 to 60 minutes more, removing to baste with the remaining glaze after 25 to 30 minutes. Warm any leftover glaze and pass it at the table.