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.
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.
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.