If both of my parents apply for the green card lottery and one is successful and gets the green card, but they do not use it to move to the US, would I (I'm not married) be eligible to get a green card and move to the US?, or would my parents have to move to the US on the green card for me to be eligible to get one?
I do not see how you can move without them. They would be the primary applicants for the DV.
For the family based green cards, does my country of origin makes difference in speeding up the process? I am from Nepal and I suppose there aren't many who are applying in this slot.
Yes. The country of birth makes a difference. Check the latest Visa Bulletin for times.