War Room is 100% necessary for persistence.
Some simple troubleshooting would be to make sure your server name in TADST is exactly the same as the one you called it in War Room. For simplicity sake I give all of my TADST profiles the same sever name so I don't have to change it with every mission I run. This is on the very top line in TADST called "Server Name."
Also, the IP address in War Room needs tone your external IP. You can find this on whatsmyip.org
When you start the mission, you should see a notification that your profile has been loaded through alivemod.com. If you see the message, persistence is working.
Regarding other players needing an account I can't say. The mission would probably persist without it as long as you have one (as admin) but obviously their profiles wouldn't persist so it would be a good idea for them to sign up.