What's with the request to update to new carrier setting?

Carrier update settings are just that, updates from your cellular carrier to make your experience on the iPhone better. They probably have made some minor adjustments to certain settings such as servers or other things. Unless you are not using that carrier, it is suggested that you run these updates.

I wouldn't really worry about the carrier updates..they are meant to correct issues. I've never had a carrier update break anything.

Not usually. However, most of the time, carrier updates show at the same time as iOS updates, so we don't always pay attention to them occurring. Every once in a while they will come up as a separate update, but as indicated, nothing to worry about.

I've only personally seen the carrier update notification when browsing on the iPhone under Settings/General/About. I believe they are automatically uploaded to the device by the carrier, then when you access Settings/General/About, you will get the update prompt. I wonder if they're only installed when accessing the phone under Settings/General/About. If so, apparently the update doesn't get installed until you do. But I know very little about this since it only happens very infrequently.

Is there where you initially saw this update prompt?

Though since I clicked on this carrier update the other day, every time I miss a call, the phone repopulates the entire list of recent calls into that window - so it looks like I've got 45 recent calls. So I delete the list and boom, next time I miss a call (since I generally don't answer my personal phone at the office) - the list is back.

Not sure where this issue is stemmed, but it surely is annoying!

A carrier update should not affect the phone's performance in that manner, however you can always try the first two steps of user troubleshooting, power the phone off/on, do a reset, holding the sleep/wake and home buttons together until you see the Apple logo and then release. The phone will reboot.