Yes.  At least he *could* have genetically dark hair.  His mother used the Occludus Potion.

Arthur's *named* father had red hair.  Arthur's biological father (a Muggleborn) had dark hair.  Arthur's mother wasn't sure whether the change was only cosmetic (magic, not genetics) so Arthur's hair was *magically* red, but could have been *genetically* dark--although this being the UK, it could have also been genetically red, just through chance.  It only mattered because Molly was also red haired, and as you say, red hair is recessive.  

Arthur's mother was not a healer or a Potion Master and didn't like to ask anyone, so she just gave Molly and Arthur a heads up.  Fortunately, it was never a problem, and the Weasleys also don't suffer the Pure Blood Blight.

Which part?  Do you mean the part where Molly talks about Arthur's mother?  

When Arthur and Molly were married, Arthur's mother confessed that Arthur was conceived with the Occludus Potion.  Apparently his biological father had dark hair and his mother worried that the potions effects might not be more than cosmetic.  If Arthur and Molly were to have a dark haired child, it would look as though Molly had been screwing around (two redheads can only produce another redhead).  That's why Molly and Arthur don't have an issue with the "Pure Blood Blight".  Arthur's biological father is a Muggleborn.  Notice that there aren't too many Pure Blood families as prolific as the Weasleys.  This is also why they were so quick to accept Sev's explanation of Harry's parentage.

The other part, that perhaps should be made clear.  Molly is pretty sure that Ginny was trying to kill herself and it wasn't an accidental OD, she's just trying to check to see what Severus thinks.

And, Severus should sit still and let himself be properly mothered, darn it.  You know she's already started on a sweater in Slytherin green and silver for him.

Does that help?