I'm sure this has been asked a million times, but it's not in the FAQ and the search function in this forum is absolutely terrible. If there are no cards making it either summer or winter, then is the state of the game neither season (possibly autumn ;p)?
Yes, if there isn't a card making a season, there is no season. Considering that all the cards that interact with seasons say "if it is Summer" or "if it is Winter," it doesn't really matter what you call it when there is no effect making it Summer or Winter because all the cards care about is that it either IS or IS NOT Summer or Winter.
I would imagine that's the case but season aren't even mention in the rules. Would it be so difficult to put all the keywords and special rules in the online errata and advanced rules, so we can get it in one place?
Actually, yes - if you are using the seasons as an example of something that should be in the rules. What "keywords" and "special rules" are you not finding?
There are no special rules for seasons. Everything about seasons is entirely card driven. You do what it says on the cards. If you have the cards, you have the rules. Effectively, a rules sheet for seasons would say "if a card says it's summer, it's summer; but it only matters that it is summer if there is another card effect that says to do something when it is summer." Not really worth it, you know?
The only keywords or special rules that are not entirely card driven that do not show up in the Core Set rule book are the Shadows mechanic and the two keywords "Joust" and "Melee." The Shadows rules are available online. And while FFG has been slow getting something for Joust and Melee online, they have put a rules insert in the packs that have the lion's share of the cards that have the keywords. If you have a card with the keyword, chances are, it came in a box with the explanation of those keywords.
Everything else (City plots, River plots, characters that become agendas, seasons, Limited Responses, etc.) are card driven. Either they are traits that don't mean anything unless a card effect specifically refers to the trait, or the instructions for what to do is entirely spelled out on the card. If you have the cards, you have all the information you need to know to use them.