There are lots of questions about the beneficiaries....I'm not a lawyer, just to my knowledge this is what i would like to put forward. PLEASE CORRECT ME IF I'M WRONG.
Every Nepali citizens living in USA since 24 June are eligible.

1)F-1: all F-1 can apply and work under TPS EAD. However, you cannot make adjustment of status to H-1, GC unless u maintain your F-1 status. So, it need to be carried together.
A2)ll F-2 and H-2: should be able to apply and can return back to their status after tps is over provided their spouse maintain their status.

3)H-1: All H-1 can apply tps too, but i don't see any EAD (from TSP) benefit for h-1 holder.

I'm still not clear about advance parole!
If a F-1 university student apply for tps and advance parole after that. He traveled back to Nepal with expired visa. When he come back to US with parole without visa stamping. Can he continue his studies in the same university? Does immigration officer inquire in port of entry and ask why didn't you apply for F-1 visa?

So my question is: How can you work through TPS EAD and still maintain your F1 at the same time?

I know lots of people does work in गास स्टेशन etc and still maintain F-1. Now they can work legally.
Going to your question, unfortunately, it's not easy. It's like in IT, getting CPT and working. unless USCIS reduce course load. I heard for some countries like Syria they have reduced course load similar to 3 credits to maintain status. Not sure about Nepal.

Ok bro...heres my situation..hope you can advise something.i was working under FULL time CPT but I had to change it to part time but my status is still active as F1 status.my question is ,can I apply for Tps EAD and work full time and still take my classes as F1 staying in current status? Sorry..too many questions but its confusing

Employment Authorization for Nepalese F-1 Nonimmigrant Students Experiencing Severe Economic Hardship as a Direct Result of the April 25, 2015 Earthquake in Nepal

What about F2 ? will they be allowed to work with TPS and EAD?

@Hereiam, Let me correct some of your understanding:

1)F-1: all F-1 can apply and work under TPS EAD. However, you cannot make adjustment of status to H-1, GC unless u maintain your F-1 status. So, it need to be carried together.
--> You can adjust your status to H1B through TPS if you have atleast Bachelor's degree. Also, if company is willing to start your GC process they can do it when you are in TPS

A2)ll F-2 and H-2: should be able to apply and can return back to their status after tps is over provided their spouse maintain their status.
--> Also they can work using TPS EAD

This is what I know understood and what make sense. the confusion arises due to taking TPS as status as we have  known normally. TPS is not status it is a emergency grant of temporary protection you have no status just a protection you will not be deported in that period and permitted to earn while in protection. it is called status because of giving EAD issue and chance of filing other status issue while in TPS. now after you clear your mind about this, your status after TPS and day after TPS expire becomes very clear. 

@ JaiPashupatinath,Absolutely u can use TPSEAD and work full time and take ur class as F-1 at the same time.

@damauli_guy: you are right in 2nd case.
About F-1: i don't want to say you are completely wrong. Yes u can work and company can file H-1 for you too. But I think there is no way for adjustment of status in TPS. So, in that case you have to go back to Nepal before your H-1 start and should return back with visa stamping. I'M SAYING ALL THIS FOR THOSE WHO DIDN'T MAINTAIN STATUS LIKE F-1 DROP OUT OF COLLEGE, ETC.
IF YOU MAINTAIN STATUS THEN IT WILL BE DIFFERENT STORY. NO NEED TO GO BACK FOR STAMPING.
THIS IS MY UNDERSTANDING IF I'M NOT WRONG.

https://www.youtube.com/watch?v=tjR7smVmfYw