IOS7 why won’t the time show up in texts

I'm afraid that's not possible in the current iOS.

But there are a few work-arounds:

• You can define a specific vibrating tone for that specific contact person which will do almost nothing (unlike the default long and hard vibrating)
• or you can disable vibrating when you enter Silent Mode and when you start having this rapid-fire conversation, just switch your iPhone to silent mode and no vibrating will happen!

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.