This chapter goes on to discuss the connections between That's So Raven and Wizards of Waverly Place. As was already shown, they take place in the same universe, which might seem to be a contradiction, but on the other hand, many different types of magic can exist in the same setting. (Just look at White Wolf!) So, without further ado.
What are psychics, then?
Psychics are people born with inherent powers, kind of like wizards, except while it shows a certain genetic trend (Raven's grandmother being psychic), it is not passed the way wizardry is. Furthermore, since Max asks if they're going to meet a psychic in "Make it Happen (Plan B)", we can assume that either wizards or non-wizards can be born psychic.
Psychic powers are magical?
There can't be said to be any non-trivial distinction between them, yes.
Can non-wizards use magic?
Yes. As shown in "Don't Have a Cow", Raven and Chelsea are going through Grandma Viv's old occultist trunk. They find, among other things, tarot cards and a spell book. This seems to imply that non-wizard magic is mostly divination, but when they cast a spell to make a wish come true, it accidentally turns them into cows. The spell they cast requires brewing of a potion, so it's possibly more like alchemy than a spell one "casts", though again the functional difference is a matter of semantics, as it's still magic. This seems to imply that they can use magic, but it's less versatile. The caveat is that this entire episode turned out to be a drawn out vision, but that implies that it would have worked. Also note that in The Suite Life of Zack and Cody, the episode "The Suite Smell of Excess" , where they most likely have a shared dream, could be an instance of telepathy among otherwise normals, and this would not be far-fetched in this universe at all. (For that matter, neither would the idea that they actually traveled to another universe, although that falls in the realm of superscience.)
Can people learn psychic powers? Or magic, for that matter?
So far, we haven't seen people practicing psychic powers, but we have seen people gain powers ("He's Got the Power") through certain celestial events. Given that there was a working spellbook, this implies that one could learn magic through that, too, although again there is probably a distinction between wizard magic and non-wizard magic in the sense that the latter is more constrained and difficult. We have seen student wizards practice magic a lot, though. This shows that while it's more powerful, it also takes a lot of practice and learning, which makes it rather unlike the psychic powers shown in the show, since those are inherent and, in the case of Raven, passive. Other psychics with more active powers are shown in "Saving Psychic Raven", though they do seem to be practicing with their powers, it could be more akin to testing them. Either way, they are limited to what they could do with what they were born with. On the other hand, since in "Leave It to Diva", Raven develops telepathy due to a cold, it implies that a psychic is born with more potential than is immediately obvious, but conditions need to bring it out. If of course, this is the case, it could also be true that everyone has the potential, given the events of "He's Got the Power". And since Grandma Viv did have a spellbook, it's likely she was trying to maximize her potential – and when we do see her, she is way more powerful than Raven, so perhaps it worked. For a short answer to this, I'm going to say "Yes, but we don't know how"
Who in their right minds would let teens on a cruise without the parents to supervise?
No clue. On the other hand, Mr. Moseby is possibly the single most responsible adult in the entire universe, though the parents wouldn't have known that going into it. Chalk this one up to the power of plot.
What happened to Dragon, the Russo's dog/dragon?
Honestly speaking, the writers didn't care enough about it. That episode was during the early days when they were testing the waters, and didn't care much about continuity.
For that matter, since when do they have a cat?
True, in the Wizards on Deck with Hannah Montana event, Justin did mention that the cat wouldn't lick Max, which is the first mention of any cat of the Russos'. However, it's not as much of a stretch to think that they have a cat we never see, after all, cats are extremely independent and known to be elusive at times. Here's a theory – they turned dragon into a cat.
How can the Leprechaun Grill justify putting a Leprechaun in a box?
Leprechauns are second-class citizens, at best. The Wizard world is notably bigoted and racist. One of Jerry's own ancestors (grandfather or great-grandfather) wrote an entire book on racist giant jokes. The fact that Alex is willing to poke a leprechaun so readily might imply that she too, is racist. On the other hand, she probably would have poked anyone had they been in the box, so it merely shows she's sadistic, which we already knew.
What's the relationship between pocket elves and elves like Tutor?
Very little. Tutor is derived more from Tolkein-esque elves, while pocket elves are closer to faeries. The faeries shown, though, are very fae-like. Flutter, when she got into a mood, only wanted to mess stuff up.
If a half-elf like Tutor and human had a kid, would it be a quarter-elf?
Kind of. Most likely they'd just say they were "part elf", the way Tutor does.
Is Alex secretly an anarchist?
I don't think it's very secret. Seriously speaking, though, the anarchy symbols in "Paint By Committee", explained as "her initial in a circle" are likely a bit of a joke on the part of the writers/producers, rather than any hint as to her political alignment. The peace signs hidden on Alex throughout season 2 are more likely in line with her political beliefs. (Which are probably libertarian.)
Speaking of politics, how did Richard Martinez get elected, and what's his party?
Presumably, after the election of an African-American, it wasn't so strange to think that a Hispanic could be elected. (Although the show started before our current president was elected, it was a clear possibility – writers do research, after all!) Demographically speaking, he's probably a Democrat.
Why aren't the Russos Catholic?
Being half Italian and half Mexican, demographically we might expect them to be. Well they might be, although we never see them go to Mass. More particularly, it's likely that being a wizard didn't mesh well with Catholics, so Jerry's (Italian) side split with the church long ago. Theresa may have decided to lay low for marrying a wizard for a similar reason, although the Church is less likely to believe such a tale these days.
In "Wizard for a Day" Alex kills a dragon in under 10 seconds. How?
Given the size of the heart, it was probably very young.