Actually, your translation is probably even better than the one given here. :) "Я думаю нет" is more like "I think, no". "Я так не думаю" also carries its sarcastic connotation just like English "I don't think so" does.
You're absolutely not right. "Я не думаю" means more "I don't think so". The sentence " I'm not using my brain"? How often do you hear it in that case?
-Он мог это сделать? Could he do it?
- Я не думаю. I don't think so.
Although Russian allows for some freedom in word order, "Я не думаю так" doesn't sound very natural. "Я так не думаю" is almost idiomatic in feel. I'm not a native Russian speaker, but that's the only way I've ever heard it spoken.