In Windows 8, if a desktop app pops up a notification in the system tray, or flashes its title bar, I don't see any notification of this at all if I'm in a modern-style app! Is there any way to enable notifications for these events?
No, this is not possible as this is a different environment, so the only solution is using a sound alert which you can hear despites being in the desktop. A lot of applications were warning you is critical have some way to play a sound, if they don't have that (in case of some kind of batch program (like converting media) you might be able to add a line in your batch script that plays a sound; for instance, with VLC player as that has command lines where you can shutdown VLC after you ran it. Otherwise you might just want to regularly check if you got a notification.
If you can get a second screen, that might be handy to put your desktop on while keeping the Modern UI on the other. Which solves your problem altogether if you can afford it.