Heat the oven to 350°. In a large bowl, toss the salt with 1 cup water until evenly moistened, like wet sand. Spread half the salt in an 8-inch baking dish, place all of the beets over the salt, then cover the beets with the remaining salt. Bake until the beets are tender, about 2 hours. Transfer the baking dish to a rack, crack open the salt, and transfer the beets to a cutting board. Peel the beets and discard the skins. Cut the yellow beets into 1⁄4-inch-thick slices and transfer to a plate to keep warm. Roughly chop the red beets, transfer to a blender along with the orange juice and port, and purée until very smooth. Press the beet purée through a fine sieve into a bowl and season the sauce with salt and pepper.
On a cutting board, remove the whole legs and whole breasts from the guinea hens and discard or save the carcass for stock. Remove the wings from the breasts and place in a roasting pan with the onions. Arrange the whole legs on one side of the pan, season with salt and pepper, and roast until beginning to brown, about 25 minutes. Place the whole breasts over the other side of the pan, season with salt and pepper, and roast until golden brown and an instant-read thermometer inserted into the legs and breasts reads 160°, about 35 minutes more. Transfer the breasts and legs to a cutting board, tent with foil, and keep warm.
Pour the stock into the roasting pan, stir to scrape up any browned bits on the bottom, and pour the stock and solids into a medium saucepan. Bring to a simmer and cook until reduced and the sauce coats the back of a spoon, about 25 minutes. Pour the sauce through a fine sieve, discard the solids, then stir in the vinegar and season the sauce with salt and pepper.
In a medium skillet, melt the butter over medium heat. Add the beans and cook, stirring, until tender, 2 to 3 minutes. Season the beans with salt and pepper and remove from the heat. Uncover the breasts and legs, pick out and discard the leg bones, and shred the leg meat into bite-size pieces. Carve each breast half from the bone and cut crosswise into 1⁄4-inch-thick slices.
Divide the beet purée among 4 serving plates and top each with the shredded leg meat, followed by the breast meat slices. Arrange the yellow beets beside the breast meat and spoon the beans over the top. Drizzle each serving with some of the sauce before serving.