All I want to do is to project a texture using a projector, and have the texture placed in the spot where the projector is aimed, not have the single texture wrapped around the whole mesh. I also want the texture to look exactly like it should, so the color should stay the same and not be transparent, unless there are transparency in the texture.
I mean, I want it to work similar to how a projector actually would, but that seems too logical.
What projector does is that it renders everything in its frustum with some material. It also pushes a projection matrix to the material shader.
To achieve an effect that the texture is projected, you need to use a shader that will render it properly. You need a Projection/* shader. I think supplied ones only do additive(light) and multiply (shadow?) rendering, but you can basically write any shader you want.
Here’s link to a question about projection shaders if you need help with that.
“color should stay the same and not be transparent, unless there are transparency in the texture”
From what you said, you will probably need alpha blending, although you also said
“I want it to work similar to how a projector actually would”
in which case the standard light shader should work (as actual projectors are actually lights, that’s why you use them in the dark room 
)
I hope this helps
If you have Unity Pro you could have the texture somewhere else with a camera facing it and use a rendered texture.
http://docs.unity3d.com/Documentation/Components/class-RenderTexture.html
This is incredibly easy in Maya… Im amazed how hard this is in Unity. Has anyone solved this yet?