1. How do I unlock more characters specifically for story mode?
Cirno, Sanae, and Meiling are the only characters with story modes in 12.3
2. How do I use custom decks in VS?
You can edit decks either through the 'deck construction' option under 'profile' in the main menu, or by pressing 'c' during character select. You can save 4 different decks for each character, per profile. The colored kanji symbol when you're selecting your character corresponds to which of the 4 decks is selected.
3. What's the best way to unlock all the cards?
To my knowledge, the only way to unlock cards is to play lots of matches, particularly with the characters that you want to unlock cards for.
However, there is a full unlock file you can download, if you'd rather skip this process:
http://koumakan.jp/games/score123.dat4. Is there a detailed guide on the weather effects somewhere?
There's a wiki for the game that has a considerable amount of detail on most things in the game, including this:
http://hisouten.koumakan.jp/http://hisouten.koumakan.jp/wiki/Weather