Sea has frozen over so I'm predicting that the Whites will just march around the wall and repeat what the Germans did in WW2 against the Maginot Line
And to address the Ice Dragon. Would seem strange for the Night King to go south without some massive play to his hand. There have been tons of wars so why choose now? Maybe it won't be an ice dragon but there is something that has prompted the NK to go south and I'd imagine it's something that men can't normally fight against such as the giants or possibly some mystical creature(s). Maybe not likely but clearly the show is going for effects over story so it wouldn't surprise me.