After spending three years towards an undergrad engineering degree in the States, I have decided that I want to live here permanently for a multitude of personal reasons. I know it is a difficult process but I am willing to give my best shot. I feel somewhat uncomfortable in mentioning that one of my primary short term life goals is to be a permanent resident here, nevertheless I am 100% sure this is what I want. Right now, I am thinking about whether I want to take up a job or go to grad school after I graduate. Since I have no preference whatsoever (I would enjoy both) and money doesn't affect my decision, I think the the best way to answer this question is to answer this "What is the fastest/easiest way to become a permanent resident in the United States (besides marriage/asylum routes)?"

I'm glad you spoke up and money is not an issue for you. In that case, the easiest route could be EB-5 visa. Just make sure, once you get your feet in, try to help other fellow Nepalese aspirants as well.

Get married to a US Citizen. 

It seems like having a PhD degree, you can apply for residency independently. But do not join PhD if the main goal is to just get a residency. But if you have a passion in some research field then it is a good way.