I think they've just had to shrink the world for convenience sake. Isn't it something like 1500 miles from Kings Landing to Winterfell? That's a journey that'll take months!
I don't quite understand this point - just because a week has passed between each episode when they're broadcast in real life doesn't mean only a week can have passed between episodes in the show. So not seeing him for a few episodes then seeing him pop up elsewhere isn't a big deal, Sam and Gilly are travelling even further but from the perspective of someone watching the boxed set it will only take them a few episodes to circumnavigate half the continent and arrive in Old Town