Are the objects in the Kuiper Belt asteroids?
This is a question that's currently being discussed by astronomers. So far, Kuiper Belt Objects (also called KBOs) are treated like they're a separate class of objects, partly to avoid having to call them asteroids or comets. What we know so far is that KBOs have different compositions than most asteroids, and different orbits than the objects traditionally called comets.
Are they comets? Some short-period comets come from the Kuiper Belt, so in that respect KBOs can be considered comets. Also, "typical" asteroids are mostly composed of rock, while "typical" comets are a mix of ice and rock. We think most KBOs are about half ice and half rock, so they may be more like comets in this way too. We think some comets "change" into asteroids as they repeatedly pass close to the sun and lose their ice, so the difference between the two classes is a little fuzzy.
Are they asteroids? You're right that some people consider KBOs to be more like asteroids than comets. Comets that come close to the Sun (including all the ones that we see in the sky with beautiful tails) have very elliptical orbits. But the KBOs have fairly circular orbits around the Sun, and most of them don't come close to the Sun at all. So if an object needs to have an icy composition and a highly elliptical orbit to be considered a comet, then KBOs are more like an "icy asteroid belt" then a group of comets.
I think what it comes down to is that our classification for comets and asteroids was based on what we knew 60 or so years ago, and it's now clear that the the Solar System is much more complicated. As we learn more about the Kuiper Belt, we'll probably come up with a different classification that works better, but for now the trend is to just refer to smaller classes of objects to avoid confusion. For example; Main Belt Asteroids, Kuiper Belt Objects, Near-Earth Asteroids, Long-Period Comets, etc., as opposed to just "asteroids" or "comets".
What is your opinion on whether an object that is in the Kuiper Belt should be called a planet if it is bigger than Pluto?
Hmmmm. I'm not sure what I would think about this one! So far, our classification for planets has relied on size: Things larger than Pluto are planets. So for the classification to make sense, I guess it would have to be considered a planet as well. If such an object was discovered, it would increase the chances that Pluto would be dropped from the list of planets, although I personally don't think there's anything wrong with calling Pluto and anything larger a planet, as long as we're consistent.
2006 Update by Karen Masters: In fact the recent discovery of Eris (previously Xena) - an object larger than Pluto and in a similar orbit did raise this question. At the IAU meeting in August 2006 Astronomers debated the formal definition of a planet and decided to exclude Pluto from the list of "classical planets" but create a new category of "dwarf planets" of which Pluto and Eris are both members (along with Sedna, and the largest member of the asteroid belt, Ceres).
If so, should Quaoar be called a planet?
Well, Quaoar is about half the size of Pluto, so I think it should be called a KBO, not a planet. I guess my personal opinion is that we should leave Pluto as a planet for two reasons: 1. Historically Pluto was considered a planet and I don't think there's a serious scientific reason to change it (changing the designation won't change what we know about it), and 2. Pluto is currently the largest KBO and is bigger than any asteroids that we know of, so (at least for now) it's a special object. Quaoar is considerably smaller than Pluto, and is about the size of large asteroids, like Ceres.
2006 Update by Karen Masters: so obviously this didn't happen - Pluto was reclassified as a dwarf planet in August 2006. There is still some debate about the size of Quaoar. At present it is not formally a member of the new "dwarf planet" class, although it may join that class when better observations confirm its size.
Another thing to keep in mind is that, in the end, this is all just terminology. Classifications are useful because they highlight similarities and differences between large numbers of objects, and when you need to talk about a large group (in this case "small objects orbiting the Sun"), it's easier to talk about general properties when they're arranged as asteroids, comets, KBOs, etc.
But even if we decided to call the KBOs asteroids, they would still be different from the Asteroid Belt asteroids (different compositions, different orbits, and different histories). So it wouldn't change how we studied the Kuiper Belt, it would just alter how our classifications were decided. The same with planets. It doesn't matter so much that Pluto is or isn't a planet: if it is considered a planet, everyone knows it's much different from the other planets. And if Pluto is just a Kuiper Belt Object, then people will still know that it's the largest KBO and most likely different from other small Kuiper Belt objects. So I think most astronomers aren't too excited about how the KBOs are classified. They'd rather learn more about KBO compositions, find out how many KBOs there are and figure out where all the Kuiper Belt Objects came from!