I am just updating this thread, so it might be helpful to other folks in future. I did travel to Nepal and came back to US using advance parole yesterday. There was no issue in Nepal whatsoever. In the immigration section in Nepal, I presented my EAD/AP combo card and the immigration officer thought it was green card and she she made a comment - "aru ko green card green huncha tara tapaiko ta red cha ta". I didn't say anything and I smiled. She then put the stamp in the passport as departed. The first time she put the stamp, stamp was out of ink, so it was not legible. She then added ink and stamped again. The second time, due to the poor quality of the ink, it messed up my schengen visa (ink got scattered in the visa page). I hope it won't be an issue as I am traveling to Europe next month.