Okay, so here is the good news.
I did the updates bit by bit... Ie: manually clicked anywhere from 50 to 100 packages and installed them bit by bit. Then, the last 399 packages, I installed them all in a bundle. And I got everything working a-ok.
I wondered if it was because I was running out of disk space on the partition. But I couldn't even check that at the time because dolphin wasn't working.
At any rate, I seem to be happily updated now. I'll keep my eye on updates in the next few weeks and report back if there are any problems.
Thanks for everyone's suggestions and advice.