Well, yesterday afternoon I ran Drive Genius from my external backup drive and used "Repair", followed by "Defrag". I also used Disk Utility to repair permissions on my internal Mavericks startup drive. Lo and behold, on-screen notifications appear to be working again in Messages. Whether or not this is a result of the things that I did, I am not certain.
However, other issues still exist, such as emoticons sometimes appearing, and sometimes not, either on my end, or on the end of the person I am messaging with.
It appears that I finally got my user picture/Avatar to stick by using the same image in Contacts, Messages and Users and Groups. I also enabled Bonjour in Messages.
Mail app is suffering even worse things, but I won't get into that in this thread.
Quite frankly, this is the worst OS update I have seen in a long time, and I am bewildered that Apple let this out the door in this state. I really like the new tabbed windows, but these other issues really dampen my enthusiasm for Mavericks. 😟