It's not a dream sequence, man. The show doesn't show that it's "mid-winter." It shows that there is snow in New Hampshire. You don't even know WHERE in New Hampshire he is. It's obviously pretty fuckin remote.
Again, even if it IS September, and the cleaner DID put his actual birthday on the fake ID - you're essentially arguing that there is no place in New Hampshire that could exist where it could snow in September, and you're arguing that based on the fact it's not winter in Albuquerque, even though the episode was actually shot IN winter, in Albuquerque.
No, according to you, Jesse's fantasy is WALT'S fantasy. Walt has fantasized Jesse having a daydream about a memory of Jesse's that, so far as I remember, was never shared with Walt.