I just realized I'd like to apply for graduate school. This gives me about 1.5 weeks to get everything together. I think I have most things in order, but I need to ask professors for letters of recommendation. Would it be appropriate to ask more than 3 professors for letters of recommendation and then decline to use them or use them at a throwaway application if more than 3 professors say yes?
Not really, no. You are asking people (with very short notice even) to write a LoR for you, which in itself is already not ideal. Then you'll just throw out that work.
Ask yourself if you'd want to spend a few hours, either taking up your free time or deprioritizing some other task, to write a letter which is not going to be used. I think you know the answer.