1.) No, you most certainly are mistaken. Doki Doki Panic and Super Mario Bros. USA (SMB2 in the States) are two entirely different games. Doki Doki Panic is the original game from which SMB2 got its engine, but the storyline and characters aren't the same in the least. Dream or not in either case, the story and character abilities hold true in the Mario canon, and it is thus recognizable as a true entry in the Mario series. Peach can float, Luigi can jump high, and Toad is speedy and pretty strong for his height--all three have retained these characteristics since SMB2.
2.) It is unclear as to whether or not Peach uses the singular Toad of SMB2 fame (among other games) or a generic Mushroom Retainer. Until Sakurai says that Toad will not be playable, there exists a technicality in the sense that Toad has yet to be officially deconfirmed. Do I think Toad will be playable? Probably not. But the possibility still exists in theory.
3.) SMB2, Wario's Woods--two games in which Toad plays the hero. You can control him in one or two of the Mario & Luigi RPGs, if I remember correctly as well. While Toad's lack of extensive roles has increased in recent years, the fact remains that Toad is present nonetheless. He may not be the pivotal character around which the storyline revolves, but he is consistently present in Mario's adventures, always lending a helping hand in whatever way he can. Toad might not be the strongest of Mario's friends, but he's certainly the most loyal.
Me too.
He looks way too good to be an AT!