IOS Messages not repeating alert notifications

Is there any reliable way to force Messages to refresh or reload the current conversation? Or make it receive messages reliably in the first place?

No. Messages is very buggy for many users and there is not a known fix at this time. I believe the most prominent client-side issues are resolved in Mavericks, but I am not running the it yet so perhaps not.

As far as I know, the best solution is to sign out of your account, log back in, and then restart Messages. For me, this fixes the ordering of messages (so they are sorted by timestamp) and generally keeps Messages running smoothly for a little while. However, the fix doesn't seem to last longer than a few days.

The source of my problem was Cydia app "Do Not Disturb".