I'm guilty of having contacted someone I greatly respect and asking "Will you mentor me?". While I honestly believe that particular individual understood the naivete in the format of my request as shown by the kind response, I would like to improve the manner in which I seek mentorship in the future.
In your examples, you offered to help improve a project of a prospective mentor with work you had already completed. If you don't know what work may help that person, would you ask directly how you may provide a benefit to him/her?
My next question is about the type of advice I seek. You suggest "[...] have gone as far as you can by yourself" before seeking mentorship. I'm not stuck, but there are a few paths I might choose. Each is a considerable time investment, and making the wrong choice would be a setback I'd really like to avoid. The conventional advice I have sought online hasn't provided a clear answer to my specific situation. What is the best way to ask someone if he/she has the time to offer advice? I don't want to start by presenting the questions and making them feel obligated to answer something that may take more time than they have available to consider.