There could be a few birds that carry it, and you wouldn't know until they were bred to each other and an all-white chick showed up. But you're right, if no white chicks ever show up it's a fairly safe bet there is no recessive white.
Correct, if all birds show reasonable amounts of black or blue, you probably do not have Dominant White. Since it is dominant, there is no way for it to hide and pop out later.
Correct, if all birds show some amount of red/yellow/brown, there should no silver in the flock. Since it is also dominant, there is no way for it to hide.
Yes.
Not-recessive-white would be CC, and also breed true.
Yes, except Dominant White is II, and not-Dominant-White is ii.
(A not-white chicken is probably CC and ii, because it is not recessive white or Dominant White. But it could be Cc ii, carrying recessive white.)
Yes, they are very pretty!
They have some interesting (complicated) genetics, too.
Here's an article that talks about what genes they've got:
https://www.feathersite.com/Poultry/CGP/Phoen/ReederRedShGenetics.html
It's a rather complicated mixture!
Oh, I think I see what must have happened.
A white chicken can ALSO carry the blue gene, without it being seen.
Black is diluted to blue, but then Dominant White or recessive white makes the chicken look white.
I've read that White Leghorns usually breed true for Dominant White (on an otherwise black chicken), but they also have blue and barring and maybe mottling as well. So if you cross them to other breeds, and interbreed the offspring, you start getting other colors (barred, blue, blue barred, etc.)
I think those diagrams are trying to show black, blue, and splash-- but they are saying "white" when they should say "splash." Maybe they think splash are dirty whites??
I also do not know why they are using a C with a superscript letter to indicate the genes, because that is NOT the standard abbreviation for that gene in chickens.
Other than the mis-labeling, that is a fine diagram for black/blue/splash.
C should be not-recessive-white, with c being recessive white (capital vs. lower case letters.) No, that diagram is not referring to either kind of white. It's just mis-labeling splash as "white" and using weird letters.
Yes, if the original whites were also carrying the blue gene (even though the breeders didn't know if was there until they got blue offspring.)
Blues come from any of the following matings:
splash x black (100% blue chicks)
blue x blue (50% blue chicks, 25% black, 25% splash)
splash x blue (50% blue chicks, 50% splash)
black x blue (50% blue chicks, 50% black)
Black can get complicated.
I would say black is mostly what happens when no other genes interfere.
Every chicken who lacks the blue gene is "black" (not-blue)
Every chicken who lacks the lavender gene is "black" (not-lavender)
Every chicken who lacks the Dominant White gene is "black" (not-Dominant-White)
And so on for not-chocolate, and not-barred, and not-recessive-white, etc.
But of course not all of those chickens actually look black.
A chicken might be lavender (two copies of the lavender gene) but not-blue and not-Dominant-White (so "black" as regards those two genes.)
Each gene has a physical location on the chromosome. That spot is called the "locus" (Latin word for "place.")
At the blue locus, the chicken can have the gene for blue or not-blue. There are only two choices. Each choice is called an "allele." Blue is one allele, not-blue is the other. A chicken gets one from each parent, for a total of two.
But there is a spot called the e-locus that has at least FIVE alleles. They all affect how the black and red are distributed on the chicken. Of course, a chicken can only have two of them at a time, because it gets one from each parent.
At the e-locus, the most dominant allele is called "Extended Black," with the abbreviation E. That gene spreads black over the whole chicken, instead of letting it have a pattern of black and red. ("Extends" the black.) Of the alleles at the e-locus, Extended Black was identified and named first, so all the others have abbreviations that begin with "e" to indicate that they are at the same locus.
The other alleles include E^R, e^Wh, e^b, e+ (and various others-- people keep finding new ones, then discovering that some "new" ones are the same as the already-known ones but others are not.)
If you want to read about the e-locus, it's discussed on both of these pages:
http://kippenjungle.nl/sellers/page3.html
http://kippenjungle.nl/sellers/page2.html
But if any single gene deserves to be called a "black" gene in chickens, it's probably Extended Black.
Of course, a chicken can have the Extended Black gene, and actually be blue all over, or have white barring, or white dots from mottling, or be all white because of Dominant White, or can have any other appearance that is caused by genes affecting black.
(Um, I did say black is complicated....

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You're welcome! Yes, I enjoy the calculator too!
If you want to play with the genes that are easier to understand in the calculator: start at the BOTTOM of the list. The top few are the ones that affect how the black & red are arranged, and they interact in ways that can be confusing. But further down the list are ones that just change the shade of black, or change the shade of red, or make white lines across the chicken, or something else that is fairly easy to understand.
(Note, I see that "mottled" does not change the picture-- a pity, since it's one of the ones you are most interested in!)
Um, I don't know for sure. I'm pretty bad at sorting out different patterns.
But wild-type hens (also called Duckwing or Black Breasted Red) usually have stippling (little black dots) on the feathers. And wild-type chicks are striped like chipmunks, too.
I'm afraid I don't know how to tell them apart.
You could post that picture and question in this thread:
https://www.backyardchickens.com/threads/the-ask-anything-to-nicalandia-thread.1509343/
I think that person is fairly good at recognizing patterns, and might be able to tell you what pattern the chicken has to go with the mottling.
Yes, that would have been great!