Seeking the night air is a literary devise you'll find a lot in novels where there is dancing. Jane Austen uses it most famously in *Emma*. Does it mean anything in the context of the Starks? Very likely.
What I find most odd is they never seem to ill. No flu, no sore throats...
Or, could Roose have been talking about Rickard Karstark? Except that doesn't make sense for a couple reasons, 1) as he talks about Lord Rickard's daughter in the past tense, as if she was dead, and Alys Karstark is very alive, unless that person at the wall is an imposter, and2) we are told that Alys Karstark was to marry Daryn Hornwood before his death. However, it's possible that Alys Karstark was a friend to Domeric and not a marriage candidate. The only other Lord Rickard we have in the north is Rickard Stark, and his only daughter that we know of is Lyanna, unless Rickard Stark has another daughter, a bastard girl somewhere in the north. Highly unlikely, however!
Maybe it's my mistake to read this as an actual race or episode of riding between Domeric and Lyanna, and perhaps it is Roose's estimation of a race between the two, but it's worded as if it happened. And if the timeline you present for Domeric being only 17-18 when he died and that was two years ago, and we know that Lyanna died approx 15 years before, and the age of 15, this all just makes no sense. And if it's just an estimation, why not compare Domeric to Brandon, another male, as opposed to a female. Odd, I tell you! But, I will over focus on this detail, and maybe someday it will all click into place for me. Or not!
While I take the wiki of ice and fire with a grain of salt, I tend to agree with their age estimate for Domeric. For him to have been fostered with Lady Dustin, then it probably happened after her marriage to Lord Dustin, and after his death, otherwise Domeric would have been fostered with Lord Dustin. In addition the times that he was a squire with the Redforts also seems to put him in his teens (late teens) at the time of his death. He's simply too young to have been racing Lyanna Stark. Which then only leaves us with one alternative, that he came into contact with Alys Karstark while he was fostered with Lady Dustin. Which makes sense because we're also told that Alys was being brought to Winterfell as a young girl to meet Robb Stark. Which was probably done to encourage a later marriage. So it's not a stretch that her father would have brought her to Lady Dustin during the time that Domeric was being fostered there, for the same purpose. I'm not terribly concerned about Roose's use of the past tense to describe Roderick's daughter as half a horse. I think that just means that he's referring to her love of horses during the time that she spent with Domeric at Barrowtown. He probably hasn't kept up with her so can't comment on whether she's still "half a horse".
Or, could Roose have been talking about Rickard Karstark? Except that doesn't make sense for a couple reasons, 1) as he talks about Lord Rickard's daughter in the past tense, as if she was dead, and Alys Karstark is very alive, unless that person at the wall is an imposter, and2) we are told that Alys Karstark was to marry Daryn Hornwood before his death. However, it's possible that Alys Karstark was a friend to Domeric and not a marriage candidate. The only other Lord Rickard we have in the north is Rickard Stark, and his only daughter that we know of is Lyanna, unless Rickard Stark has another daughter, a bastard girl somewhere in the north. Highly unlikely, however!
Maybe it's my mistake to read this as an actual race or episode of riding between Domeric and Lyanna, and perhaps it is Roose's estimation of a race between the two, but it's worded as if it happened. And if the timeline you present for Domeric being only 17-18 when he died and that was two years ago, and we know that Lyanna died approx 15 years before, and the age of 15, this all just makes no sense. And if it's just an estimation, why not compare Domeric to Brandon, another male, as opposed to a female. Odd, I tell you! But, I will over focus on this detail, and maybe someday it will all click into place for me. Or not!
While I take the wiki of ice and fire with a grain of salt, I tend to agree with their age estimate for Domeric. For him to have been fostered with Lady Dustin, then it probably happened after her marriage to Lord Dustin, and after his death, otherwise Domeric would have been fostered with Lord Dustin. In addition the times that he was a squire with the Redforts also seems to put him in his teens (late teens) at the time of his death. He's simply too young to have been racing Lyanna Stark. Which then only leaves us with one alternative, that he came into contact with Alys Karstark while he was fostered with Lady Dustin. Which makes sense because we're also told that Alys was being brought to Winterfell as a young girl to meet Robb Stark. Which was probably done to encourage a later marriage. So it's not a stretch that her father would have brought her to Lady Dustin during the time that Domeric was being fostered there, for the same purpose. I'm not terribly concerned about Roose's use of the past tense to describe Roderick's daughter as half a horse. I think that just means that he's referring to her love of horses during the time that she spent with Domeric at Barrowtown. He probably hasn't kept up with her so can't comment on whether she's still "half a horse".
I probably am overthinking this whole deal, but I think the past tense use is very odd. Roose is very deliberate in his word choice, and I think GRRM does that for a reason. Still, it makes some sense if Roose is talking about Lord Rickard (Karstark's) daughter, Alys. We are never told that Alys met Domeric, but it's certainly possible that they did, as you have described. We are also never told she is a particularly good rider, but she did manage to get from Karhold to Castle Black ahead of Cregan and his men, and that could indicate that she is rides well. I do think there is the potential of a plot afoot with Alys and she is trying to manipulate Jon, which makes her story a bit suspect.
On the other hand, we are told Lyanna Stark is a great rider. Barbrey Dustin refer's to her and Brandon as centaur's, and Roose's mystery "Not even Lord Rickard's daughter could outrace him, and that one was half a horse herself" could very well be in reference to Lyanna, whom is already referred to as a centaur, half man half horse. It could be Alys, but we are never told of her riding skill, and she and her horse arrive at Castle Black in rather poor condition, which could be the cold, but could also show a lack of skill as a rider.
The only other "half a horse" idea I can think of is House Caswell, with a black centaur on their sigil, but we are never told there could be a Lord Rickard in the mix, although I think the current Lord Caswell has not been referred to with a first name in the text, nor is there a daughter ever mentioned, so this is highly unlikely. And this is a family from the reach, so not likely to play a part in Roose's reference.
Roose's use of "Lord Rickard" is pretty vague in this case. Two other times in the text Roose uses the term, both from Theon/Reek's POV, and once is in regards to Karstark and once in in regards to Stark, and both in the same chapter.
Karstark: "After the scratch the Young Wolf gave Lord Rickard, that may be somewhat less true than formerly. ADWD-Reek III
Stark: "A year later this same wench had the impudence to turn up at the Dreadfort with a squalling, red-faced monster that she claimed was my own get. I should've had the mother whipped and thrown her child down a well … but the babe did have my eyes. She told me that when her dead husband's brother saw those eyes, he beat her bloody and drove her from the mill. That annoyed me, so I gave her the mill and had the brother's tongue cut out, to make certain he did not go running to Winterfell with tales that might disturb Lord Rickard." ADWD-Reek III
Much of the text is vague, which certainly add's to the mystery. I am unconvinced either way, but I just think it's a bit odd that Roose would refer to Alys Karstark as half a horse, and in the past tense, but it certainly does fit the idea of the timeline as we have it. There are many things I find odd in the text that might turn out to mean nothing at all!
Their father understood as well. "You want no pup for yourself, Jon?" he asked softly.
Much of the text is vague, which certainly add's to the mystery. I am unconvinced either way, but I just think it's a bit odd that Roose would refer to Alys Karstark as half a horse, and in the past tense, but it certainly does fit the idea of the timeline as we have it. There are many things I find odd in the text that might turn out to mean nothing at all!
Maybe I need to look at this as a hint that Domeric had Stark blood, possibly from Brandon, as I had initially thought years ago, and this was Roose's sly connection to the family without naming Braondon specifically. It could be a reason for Roose to feel he must seek vengeance from the Stark's. I can't help by think Roose has been making sly inroads in the north for many, many years. A quiet peaceful rule on the surface but much turmoil and plotting and undermining under the surface.
Or this could be as simple as Roose comparing great riders of different generations, such as we much do with sports stars or celebrities. Who is the greater heavyweight boxer, Jack Dempsey or Mohammad Ali? They never faced each other, but were both great, but one would have to beat the other if they could have faced off at the height of their respective careers.
Or this could be a reference to Alys Karstark, as freyfamilyreunion pointed out.
Several ways to look at this comment by Roose, and probably none of them will affect the end of the story, and it's another think I can stop fixating on!
Their father understood as well. "You want no pup for yourself, Jon?" he asked softly.