She can not remain in dual status, either H4 or TPS. Since her TPS is approved, you should know that her current status is TPS and once TPS expires ( or you may cancel TPS), she will be back to her H4 status again (provided your H1B is still valid).
Now travelling in TPS or H4:
Since USCIS knows that her current status is under TPS, she can travel to Nepal under AP through TPS. I am not sure if she can still be able to travel in H4. But I am sure that if she already used her EAD through TPS, she need to travel with AP through TPS otherwise she will have problem in POE.
Getting AP through TPS
It will take around 90 days to get AP through TPS if you apply now. If you need AP soon, schedule infopass with USCIS and ask for expedite processing. You should have a valid reason why you are going to Nepal under emergency.
I-140 processing:
I do not know how long it takes time to approve I-140 under normal processing (PP will take 15 days only with extra cost). If you I-140 is approved, you can file AP through I-485 for both of you and can travel to Nepal after that. I do not know if you can get AP before this Dashain.
Your questions:
Q. She is starting her new job today with TPS EAD. We are planning to go to Nepal this Dasain for a month. Do you guys know if working with TPS EAD will revoke her H4 status, and will it be problem while coming back even though she has valid H4 visa?
A. Once she uses TPS with EAD, she is active in TPS and H4 status is on HOLD (not revoked). Since H4 is dependent status to your H1B, as long as you H1B is valid, her H4 remains valid too. She can not travel to Nepal with H4, she need AP through TPS.
Q. If we apply advanced parole, and come with it, would her H4 be still revoked and would cause issue during GC processing ?
A. NO. TPS and GC is two different things. GC processing will have an issue only if you guys are in out-of-status previously.
All these are my personal understanding and always recommend to consult Nepali Lawyer who are expert in TPS and immigration issue before taking any decision.