1. Soak wood chips in water 1 hour; drain.
2. To prepare shrimp, combine paprika and the next 9 ingredients (through red pepper). Combine olive oil and shrimp in a bowl, and toss to coat. Sprinkle shrimp with rub, and toss to coat. Cover and refrigerate 30 minutes.
3. To prepare rémoulade, combine mayonnaise and the next 10 ingredients (through garlic) in a medium bowl. Remove yolks from eggs; discard yolks or reserve for another use. Finely chop egg whites, and stir into mayonnaise mixture. Cover; chill.
4. Remove grill rack, and set aside. Prepare the grill for indirect grilling, heating one side to high and leaving one side with no heat. Pierce the bottom of a disposable aluminum foil pan several times with the tip of a knife. Place pan on heat element on heated side of grill; add wood chips to pan. Let wood chips stand for 15 minutes or until smoking; reduce heat to medium. Coat grill rack with cooking spray; place on grill. Place shrimp in a grilling skillet; place skillet on grill rack over unheated side. Cover and grill 15 minutes or until shrimp are done, stirring after 7 minutes. Serve with rémoulade.