I'd contact the developers about the Online Bible app. There's really not a lot of basic stuff that changed from 10.8 to 10.9. Of course, if it's a 'ported' (from Windows) app, the market share may not be big enough for them to bother with even minor changes.
How do you have messages in Mail sorted. I use "Date Received" 99.99371% of the time, at least on days ending in "y". But I also have Mailboxes with absolutely nothing selected when I open it. Generally, the only ones that are even selected are ones that I access daily and delete/read. There may be a setting in the prefs that affects this behavior; the only one I see id Viewing->View conversations->Show most recent message at the top. That seems to apply to conversations, of course, not single messages.
I assume a "Widget" is a Dashboard thing? There is still a Dashboard, it's an app, should be in the Applications folder. I'd suggest starting up Dashboard and see if you can find/run/stop/delete that widget. You might also want to check your Startup items list... OTOH, your "HP" may not work correctly with that thing turned OFF. And you want to be sure that all hardware has updated any required drivers for
Mavericks.
Not everything uses Notifications, yet. I still like Growl better, anyway. But if you'll look in your System Prefs and select Notifications, you'll see the apps that can and do use it. To completely stop Notifications for anything that uses them, select it in the "In Notification Center" and click the "None" alert style plus UN-check the "Show in Notification Center" checkbox.
Since this is coming from Messages, you might want to look at its Prefs, although I don't see any specific Notifications related choices.
