"What you CANNOT do in TPS EAD but do in OPT EAD?
- apply H1B visa, and have Change of Status from F1 to H1B without leaving US.
- apply Green Card (employer sponsored), and Change of Status to Green Card without leaving US."

The above info is incorrect. You CAN apply for H1B or Green Card from a TPS EAD as long as a company sponsors you after meeting DOL (Department of Labor) criteria. The only places that will not sponsor an H1B/GC, or even hire you for that matter, from TPS EAD are government/defense related institutions that require US citizenship and security clearance (NASA, Lockheed Martin, etc. Sometimes there may be positions that do not require security clearance but most of the times they will).
Someone on sajha also posted a spreadsheet some time back showing prior status of green card applicants and a bunch of them were on TPS. Not Nepalis necessarily but from other TPS designated countries.

Also, as for adjustment of status, you don't have to leave the country if you entered legally and have NOT been out of status. Check out my posts in the below thread for some subtleties regarding being out of status.
http://sajha.com/sajha/html/index.cfm?forum=55&ThreadID=117642

I am not an immigration expert but you should speak to one before acting upon my or any one else's advice on this forum. Good luck.