Welcome to the Cycle Diet! My answer to you is it depends on the nut. Some nuts have a better protein profile than others. You're looking at about 4 grams of protein per oz of most nuts, peanuts have about 5 grams of protein per oz, but they also have more saturated fats.
If you are just starting out on the Cycle Diet I recommend either walnuts or almonds due to their great fat profile and omega 3 content. Peanut butter is okay as long as it's the natural kind that separates.
Regarding amounts it really depends if you are trying to limit calories, since most nuts are calorie dense. I like to limit nuts to a serving size at around 1 oz or about 20 almonds or 10 English walnut halves or about 1/4 cup. If you're just adding them to your cereal or salad than just use a few as a garnishment. If you have a calculator handy here's the break down on calories. There are 16 Tablespoons in 1 cup if your going to estimate.