Corrected entry: This episode takes place during the holidays and Robin is missing her family. At the very end of the episode Robin and Marshall are sitting outside - Robin is eating an ice cream cone and Marshall is drinking an icee, and they are both wearing T-shirts. Behind them are poinsettia plants and they mention Christmas in the episode. They live in New York. It snows in New York during the holidays or is at least cold. They would not be wearing T-shirts sitting outside on the stoop eating ice cream.
Corrected entry: There are two problems with the scene in which Heather enters the store with the briefcase. One is debatable, but one is not. All stores, for safety reasons (most notably, if a fire were to break out) should have doors that open outwards, away from the store. It's a law in most areas of the US. The other thing is that Ted says, "That's a "push," not a "pull."" That means that she attempted to pull it, but if it were a "push" door, there would be no handle to pull, meaning there is nothing there for anyone to pull on.
Correction: No they are both debatable. There is nothing wrong with having handles on both sides of the door. No mistake here. As for the 'safety' concern. You said 'most areas' of the USA have this rule. Can you tell me that every store in NYC abides by this rule? I would imagine there would be a lot of stores that don't, this store for example. No mistake here.
Correction: The episode makes repeated references to the fact that they are from colder areas (Canada and Minnesota). As such, neither find the winter climate in New York particularly cold, something that is exaggerated to the point of them behaving as if it is summer weather. It seems you missed one of the key jokes in the episode.