1. To live on the mountain is any location on the mountain. To live at the top of the mountain is very specific about being near the peak.
2. "Scrap* paper" is fine. Often scrap paper is used to make notes, practice writing on etc. "Useless paper" is less appropriate and would only be used in specific situations. "Waste paper" is what you throw away.
3. "I want to expose your lies" means I want to make the information publicly available. "I want to disclose your lies" would really need a target audience to be specified, for example "I want to disclose your lies to your employer". "Expose" is more natural in most situations, I think.