Why is House Oakheart known for their hatred of the Dornish? I get them joining in on wars that the Reach has with Dorne, and an Oakheart died alongside Daeron the Young dragon, but they’re so far north west on the map. Shouldn’t they have a tradition of fighting westermen and defending the Reach from Crakehall invasions and whatnot?

They should indeed. Ironborn too, given the proximity of the Shield Islands. 

But as a principal House of the Reach, they’re going to be first and foremost in the ranks of the Gardener Kings’ lieutenants. So I wouldn’t be surprised if Egderran Oakheart was a general in the armies of Garth the Hammer of the Dornish, Garth Goldenhand, or Garth the Painter, since he seems to have been victorious. Alester Oakheart has a strong Roland vibe to him, so he might have been a leader in Greydon Gardener’s failed invasion, or the invasions of the Reach under Gyles III or Garth Xth.