To prepare salmon, combine first 5 ingredients in a small bowl. Sprinkle one-third of dill mixture in bottom of a 13 x 9-inch baking dish. Arrange 1 salmon half, skin-side down, on dill mixture. Combine one-third of dill mixture and beets in a small bowl; spread over salmon. Top with remaining salmon half, skin-side up. Spread remaining dill mixture evenly over salmon. Cover loosely with plastic wrap. Place a cast-iron skillet or other heavy object on top of salmon to weigh it down; refrigerate 24 hours.
Remove skillet; set aside. Uncover salmon; carefully turn salmon stack over. Cover loosely with plastic wrap. Place skillet on top of salmon, and refrigerate 24 hours. Repeat procedure one more time.
To prepare salad, place cucumbers in a colander; sprinkle with 1 tablespoon salt. Toss well. Drain in sink 1 hour. Place cucumbers on several layers of paper towels; cover with additional paper towels. Let stand 5 minutes, pressing down occasionally. Combine cucumbers, vinegar, and remaining ingredients except dill sprigs. Cover and chill 1 hour or up to 4 hours, stirring occasionally; drain well.
Scrape off and discard beet and salt mixtures from salmon. Discard liquid. Cut salmon into (1/8-inch-thick) slices; discard skin. Serve with salad. Garnish with dill sprigs, if desired.