I have had this problem. Please take time to report it by filling out a bugzilla request.
1. When I went to edit an existing distribution list in the admin page, I got a member already exists error. At the end, I have to delete the distribution list and redo it again. Is there anyone who faces the same problem?
Yes, they will. The reason the search returns everyone is because there is a "." in the domain name.
2. When I search a "." in the distribution list, it returns all the names created in the server. But it includes the aliases as well. Will the person receive two emails if I sent to that list?
Currently (as far as I know) you can only add items in your domain to your distro list. Also, currently the search only searches name and e-mail, not the comments. . .once again, bugzilla feature.
3. There must be a way to search other by display name or email address (to put the items in the distribution list. Is there? Can I search other fields? If so, what fields?