How would the politics of the North change during the War of the Kings and after if Roose’s trueborn son Domeric had survived? Thats assuming Roose succeeds in removing the Starks from power.

Well, Domeric’s survival might change a lot of things. To begin with, we don’t know if Domeric would have gone south or stayed at the Dreadfort – if he’s there at the Dreadfort, the Hornwood Affair doesn’t happen, and the North rallies against the Ironborn invasion, eliminating the conditions for Roose’s switching sides. 

Even if he’s gone south, it’s quite possible that Domeric might have been with Robb’s army rather than Roose’s – lots of other Northern sons were in Robb’s group of companions, there’s the infantry/cavalry split among the Bolton forces that requires two commanders, etc. Roose would be much more hesitant to betray Robb if he risked losing his heir in the process.