There's a distinction between language and reality. Language consists of verbal behaviors and concepts which may or may not have a correspondence to reality. So having a word doesn't mean that there's a corresponding object. Previous posters have mentioned classic examples like unicorns or even the obviously logical contradiction of square circles.
But there's something else here that you touch upon which I think has some validity. Is nothing really the complete absence of anything? We can see many examples which seem to contradict this, all of which revolve around the fact that these nothings seem to do some real work.
For instance, you bring up 0. 0 is supposed to be nothing, no quantity, yet its role in math is significant, and some consider it one of the most important developments in mathematics, possibly being the thing that made positional notation practical and with it, the relative ease of calculation that we have today. It's hard to imagine something not existing (having unreality) while simultaneously having this very real effect.
Or take the more mundane example of the space inside a cup. A cup is hollow inside, hence most people would think there's nothing there. Yet, it's this nothing that allows the cup to hold fluid, and which gives it its use. Without this nothing -- with a "filled cup" we'd have a useless chunk of matter.
So really, maybe nothing isn't nothing? If something exists in some level -- even as a contradictory idea -- it can have some impact on reality. Unicorns don't exist, but the idea of them has caused paintings, shows, etc... to feature them. That's certainly something. Square circles don't exist, yet we write about them, discuss them, and their analogies involve our discourse on contradictions. That's certainly something.
So we use nothing in a relative sense, not in an absolute sense.