Let’s say the White Walkers don’t exist and Stannis defeats the Boltons but he doesn’t have the strength to project his power southwards. How long can Stannis rule as King of the Iron Throne in the North? When he dies would the Northmen recognize Shireen as queen, crown Rickon/Bran as King in the North, or bend the knee to Dany?

I think the key factor here is whether he’s got Rickon and Bran. If Stannis is seen as the restorer of the Starks, he’s got a lot more credit than he would otherwise. If not, he’s got 1,500 men in a country with 17,000 men under arms who are going to evict him eventually. 

Long-term, I think the North is going to be very, very hesitant to ever rejoin the Seven Kingdoms, at least not on the original terms. If Dany survives the Second Battle for the Dawn with dragons, they might bend the knee, but that’s as far as it goes.