To my understanding the transmission (02 Honda cbr600f4i) is in the crankcase, and is the normal engine oil the lubricant for the transmission? If so what weight of oil is good for both engine and transmission, and any favorite brands
Yes and check your owners manual for recommended weight. Probably 10W40. To many brands to chose from but I like the Gold Spectrum and Silkolene. Next question is going to be whether to use synthetic or not. My 2 cents on that is, use a blinded oil for the first few k and switch to a full synthetic true motorcycle oil. Not all automotive oils are made for the wet clutch your bike has.