"By the way after staying in america for 5+ years, don't these fake asylees get american citizenship? Or they become asylee in america for the rest of their life?"

It's a choice they make. They can retain their Nepali citizenship and US Permanent Residency for rest of their lives. Why does it even matter what they do with their private lives?

"If they want to travel to Nepal when they are asylee in america, does US allow to do that? Or these fake asylees go to India and sneak into Nepal?"
I think this is true. But there's gotta be time limit to when they are not allowed to trave back to Nepal. It cannot be rest of their lives. Once granted US citizenship (if they choose to take up one), I don't believe they would be bound by such rules. All US citizens would and should have the same rights. Having said that, if their fraudulent asylum application is ever discovered by the authorities, they can revoke their citizenships and subject them to criminal proceedings and of course deportation. I don't think there will be stature of limitation in such cases...