Here are some questions I got and my answers:

1. Does it mean that we have to drag&drop the items only in one way A-to-B?

One way from A to B is fine. If you think doing drag&drop in both diretions is easier, you can do that too.

2. Does it mean that we have to remove the items from A and add them to B, or we have to implement just copying the items List A to List B?

Either moving or copying is fine.

3. Do we have to show only the scripting part of the code or entire html page?

Just the script part.