Snow also could not ever become the heir to the Starks because he is now a Night Watchman. Part of their oath is that they give up all claims to titles and cannot father children and it is an oath for life.
A couple questions related to that is how can Jaime Lannister reconcile his oath with the Kingsguard which is similar to the Night Watch (give up all titles and claims and can't father children which he obviously has done in secret) as well as Lord Mormount who I believe is Jonah's father but how could he have done so? It's odd as well because Jaime was first in line for the Lannister title so why would he have given it up? Why would Lord Mormount have done the same with the Night Watch and who is Lord of the Mormount House now?
The HBO site has some great background that isn't spoiler oriented (though some folks think any information is a spoiler), it just explains the world and basically fills in the blanks the book I assume (just started reading) can detail. For instance it explains the Night Watch, the Kingsguard, the City Watch, the Maesters, etc. and what their obligations are. It also discusses how the world came to be up until this point in the story (how the Targareans took and lost power, who all the Houses are and how they are connected, interactive map, etc.) For instance, in the book when a character refers to a name or an event it can take a paragraph to explain context but in the show they have to keep going and you just hear a name of someone without knowing what it is. My guess is they will continue to have scenes like when the daughter is in the throne room with the teacher asking her questions on previous kings in order to give background.
The intricacy of the story is mind boggling.