You'll never get Albino offspring in the first cross with a non-Albino as Albino is a recessive trait. Also I have found that the "True" Koi trait is also recessive therefore you won't see any Albino Kois until F2. You will see non-Albino/ non-Koi Lyretails in F1.**Recently I have seen non-Koi fish labeled as Koi (these fish were Painted and Peppermints).
A dominant trait such as Lyretail, Hifin or Plumetail will show up in the F1 (first cross) generation when one of these is bred to a fish without the trait. A recessive trait such as Albino and Koi will show none of these in F1. I'm not a geneticist so bare with me, I hope I was clear.
No, the difference between a "True" Koi and some of these other fish being labeled Koi is a Koi has a certain pattern which doesn't change much from fry to adult (Red & White not Red & Black). A Peppermint/ Piebald has many different and changable patterns.
I have a tank of albino koi regular fin and I will introduce the non albino lyretail female. Sounds like I should get some koi lyretail but probably no albinos in the first spawning. If I get a non albino koi lyretail female and then pair with a regular fin albino koi I should get some albino koi lyretails from this pairing.