Hello,
I am about to graduate this December and want to invite my parents over for my graduation. I have the invitation letter from the University ready and was wondering how early should it reach to Nepal. I heard that currently there is 1 to 2 months wait time to get the visa interview so it seems like they need to get the date asap. I also heard that the invitation letter is required to get the date. Can I just send the scanned copy of the invitation letter to Nepal while they get the date, and let them present the original during the interview date.

Yup, the scanned copy of the invitation letter will work. No need of sending the Original one as far I know.
I know the embassy does not require any other documentation besides the invitation letter, appointment confirmation letter and I think receipt of payment, I suggest you to send a copy of your status (May be copy of i-20, showing you are graduating soon) just to be in safe side.
Hope this works.