F1 versus F2 generation confusion (Genetics)

I am in serious need of correction or confirmation please! I am going to throw out some random facts and please let me know if I am accurate or not. Thanks for helping!

F1 hybrid can only be created by using two separate parents from inbred lines or P1. F1 are uniform, showing a new phenotype without variation. Not even once in a while will you see a second phenotype show in f1. If you cross an f1 to a separate types of F1 you get an F2. If you cross an F1 and an F2 of seperate lines and you get F3. If you cross an F1 with a P1 from different lineage from the F1's parents, you don't get an F1 as a result of it but an F2. F1 is the only type that shows improved fitness, while their offspring loses fitness with each generation. If you take an F1 hybrid and cross it back to one of the parents or plant from that line you get BC1. If you take an F1 hybrid and cross it to a BC1 from seperate lineages you get an F2. If you take an F1 hybrid and cross it to a BC1 of the same lineage you get BC2. If you cross an F1 hybrid to itself you get F2.