1. Why do we build models?
I build models for a few different reasons. I'm a military history buff and so I enjoy anything I can do related to this topic (for instance, I'm always on the lookout for tours of military museums/ships/bases, I watch the History Channel constantly, I play military video games, I build military vehicles, etc.). It's very difficult to find a BM-13 multiple rocket launcher or an F-86D in Cleveland Ohio and just searching for pictures can get kind old quick, so building models allows me to take a tour of these historical masterpieces without having to leave my house. Also, I find that I can get a better understanding of the technical specifications and sheer power of these vehicles, their importance in history, and the fear they instilled in their enemies during combat. In short, it's a way that I can personally connect with the past. Aside from that, it's a stress reliever, it helps me center myself, and it's also a good way to keep my coordination abilities.
2. Are we misunderstood to what level by non-modelers?
I do think we're a bit misunderstood. From what I've experienced, a lot of people assume modelers are skinny little dweebs who have no friends and can't go out in the sun. Then I show them pictures of my work and they usually change their opinions after that.

I don't know where this bias came from, but it's pretty easy to change people's minds when you show them what we can really do!
3. What do you do with your finished model?
I display them in my room. I'm in college now so I can't really do much else. When I finished my sixth (or so) model, I cleared all the books off of my bookshelves and that's where they rest now. However, these shelves aren't the best for display, as they are poorly lit, don't keep out the dust, and there isn't much room for many more models. When I'm a little less mobile, I hope to buy or build myself a large shelving complex with lighting and glass doors to do justice to my work.
4. Can our building hobby be considered art, therefor entitling our breed as amateur artists?
Absolutely modeling is art! Many people consider it just a hobby, but isn't painting a hobby as well? The fact of the matter is that when one builds a model--even one from a kit--he or she puts an immeasurable amount of their own skills, creativity, knowledge, and imagination into the project. I dare someone to watch a modeler build their kit and then say that the modeler is not an artist.