I believe the red hair was from the Spencer Family.
Red hair is recessive. So he gets it from both sides of his family.
If the rumours about Hewitt are without foundation, they would have been scotched by the BRF. Discreet DNA testing and some subtly worded press statements would have sorted it out years ago.
Truth is that the BRF either don't want to deny of the rumours, or they cannot deny of the rumours. It's not rocket science.
I find it very dubious that a DNA test wasn't done by the press already. To me, it seems like the easiest thing to do. Hewitt is easily accessible. Harry isn't, but he has the clubbing, the snogging girls, the engagements and the military carrier. At some point, viable material is bound to appear. The British papers are ruthless, far from incompetent. IMO, a test (several, in fact) by the media happened but it failed their expectations, and Harry is the prince's kid.
As to why the BRF has never commented upon it... Well, they seem to operate under the premise that it would only be something if they addressed it, which I agree with. Completely ignoring is the best way to say this whole business is rubbish. Harry kept all of his titles and is still third to the throne, the BRF is endorsing him. Furthermore, if they made the tests and showed them to the press, there would be two problems: the first, that they actually considered it (with a plus of doubting Diana and mud sling at her after she's dead), and second, give precedence for the press that if they're bullied enough, the BRF will give in.