You unlock new characters by spending currency. For the larger ones, I think you typically have to unlock it through a different related character. But larger doesn't always mean better, you may find you prefer the four panel versions.
It's been a long time since my save game got deleted after beating the game, but I'm pretty sure the other mode were unlockable quizes. Nothing you need to concern yourself over unless you read japanese.
I think it's possible to cheat online, but generally you're probably playing against some kid that has spent the last few years playing this game and reduced cheap+spam tactics to an art form. People bitch endlessly about this fact, so you can either get used to it by not worrying so much just about winning, or you can do like everybody else and set up your own matches against players that you know.