Try it again:

I will ask the copy of the agreement. My questions are:
A. If there are such wordings in the agreement about EAD/AOS,

1. Whether I am actually working on EAD even if Company V has no copy of my EAD? In other words, my EAD is activated automatically?
-No. As long as your company does not withdraw the approved H1, your status will be H1 until the H1 I-94 expires no matter whether the company has your EAD or not. So, just ask the company not to withdraw your H1.

2. Whether I can ask Company V to maintain H1-B?
B. If there are not such wordings in the agreement about EAD/AOS,

1. Whether Company V can still put me as EAD status just because I got EAD, even if Company V has no copy of my EAD?
-The company needs to have your legal work permit such as H1 or EAD. It does not matter to you to show your EAD to the company.

2. If B-1 is true, what I can do to maintain H1-B?
-Just ask the company not to withdraw the approved H1. I am sure what the company consider is they do not want to spend money to extend your H1, so keeping your currently approved H1 does not hurt them anything.