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Not dumb questions at all. I'll do my best to answer:
Yes, yo can play the Dragon's Demand campaign right out of the box with just core. It's a full adventure path, albeit shorter than most around here are used to, but you don't have any expectations coming in, so no worries.
Curse of the Crimson Throne adds a bunch of new cards and is a campaign more akin to what the previous adventure paths were, and more than twice as long as Dragon's Demand. If you enjoyed playing through DD, then getting Curse gives you a much more complete and in-depth experience.
The Character decks serve two major purposes. First, as you suggest, they give you access to new characters and cards that are good for the respective classes, which could spice up an adventure for your 4th or fifth play through. Second, they are needed if you want to participate in the organized play aspect of the game, which utilizes cards in existing APs to tell a new story, and each player will utilize their own class deck and cards, which means that players can swap in and out as they want since their deck is independent of the adventure itself. I'd recommend them only after you have both Core and Curse and know that you're going to be playing this game a lot.
Also, to give you my personal experience, my wife and I have been playing the ACG since the first set, and it is hands-down our most played game. I actually proposed to her by homebrewing an adventure for her birthday and playing through end with giving her the ring as a loot from winning. So, long story short, I recommend it.