Combine the crushed red pepper and paprika in large bowl and stir in 2 tablespoons of hot water and let stand for 5 minutes. Add the yogurt, olive oil, red wine vinegar, tomato paste, 2 teaspoons salt, and 1 teaspoon black pepper to spice mixture in bowl; whisk to combine. Stir in garlic and lemon juice, then chicken. Cover and chill at least 1 hour.
Prepare grill over medium-high heat. Thread chicken pieces on skewers, dividing equally. Discard marinade in bowl. Sprinkle each skewer with salt and pepper.
Grill chicken until golden brown and cooked through, turning skewers occasionally, 10 to 12 minutes total. Transfer skewers to platter.
Serve with tzatziki and extra lemon wedges