OK here comes a short answer to a long question...
Cost index relates fuel costs against trip time costs. So a slightly faster climb speed will be posited against a slightly reduced rate of climb. This is - as you correctly assumed, achieved by the Cost Index. The BEST Rate of Climb is achieved at Cost Index 0. (You don't need a spoon feed here - think about it).
Derated (reduced) climb will save a little engine life, but costs about 3-5% on fuel. So if you are short on the liquids from the getgo - climb at full climb power and save 200-300 kgs (local mileage will vary depending on type).
MG