Whisk oil, vinegar, shallot, paprika, 1 tsp. salt, the oregano, and pepper in a medium bowl. Chill 1/2 cup of dressing. Add chicken to remaining dressing, turn to coat, and chill, covered, at least 1 1/2 hours or overnight. Let chicken and reserved dressing stand at room temperature about 30 minutes.
Meanwhile, heat grill to medium (350° to 450°). Transfer chicken to a plate. Add corn and mushrooms to marinade and brush to coat. Grill vegetables and chicken, turning once, until vegetables are streaked brown and chicken is no longer pink in center, 15 to 20 minutes for corn and 12 to 18 minutes for chicken and mushrooms. Transfer to a board as done.
Cut warm corn kernels from cobs into a large bowl. Cut chicken and mushrooms into strips; then add to bowl, along with spinach, onions, and reserved dressing, and toss to coat. Season with salt if you like.