because the gene for red hair is recessive...(genes can be dominant and recessive over each other)
it's likely to disappear slowly, just like blond hair. Or blue eyes
This is somewhat true.
Yes, there are recessive and dominant genes, but recessive genes don't necessarily disappear just because there's a dominant gene. In terms of genetics, dominant just means a gene can express itself without matching with another. It can override the instructions of a recessive gene if it's paired with one. As long children inherit recessive genes from their parents, whether it expresses itself in the phenotype or not, and their children get it from them, recessive genes don't disappear and will, eventually, be matched up with it's complementary.
The gene for red hair is considered to be recessive. However, there is evidence that it may possess "incomplete dominance," meaning it essentially compromises with whatever genes it's matched up with and tries to express itself regardless of what color the hair really is. This explains why there are so many shades of red hair (strawberry blondes like Nicole Kidman, for example).