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It's not that it was "created" from birchen. The black copper pattern is based on what is called the "birchen" gene -- notated as ER. "Birchen" is just the label that is attached to that gene.
The main difference between what we think of as "birchen" birds and what we think of as "black copper" birds is the silver gene vs. the gold gene. In addition the black copper also has the mahogany gene, which the birchen should not. But both have the same "E" allele, which is ER.
Wheaten is based on the "wheaten" gene, notated as EWh. Wheaten birds also have the gold gene (unless they are silver wheatens, which you don't often have to worry about) -- so if you cross a wheaten bird with a birchen bird, you'll get a birchen bird that is carrying gold. And that "mixed" bird does indeed look kind of like a black copper. However, it isn't a "real" black copper because it will usually not have any mahogany (some wheatens do have mahogany, but really good wheatens won't), and it will only have one copy of the gold gene. These two things together -- only one gold gene and no mahogany -- will make the bird look much paler an more washed out than a "real" black copper.
Also, if I get a Wheaten from Black Coppers, is it okay at all to introduce it to the true Wheaten breeding pen? What genetically is the difference?
See above. Wheatens that come from black coppers are likely to have mahogany, and also to have extra melanizing genes that the black coppers need but the wheatens shouldn't have. You CAN introduce them to a breeding program -- in fact, I'm doing that right now -- but you'll have to breed out the unwanted genes.