MacOS – Can the alert sound in Mountain Lion’s notification center be customized

There might be another app that will work, eNotify. While I haven't tried it there is a lite version you can use to see if it works for you. This is along the think different solution. Send the email to a special email account and get annoying alerts.

eNotify provides direct local notification for incoming messages along with custom email alert sounds for different accounts, senders, subjects, and recipient addresses. Easily silence background email noise and stay aware of important message traffic! In addition, the message sender and subject can be shown via local notice, so you'll always know when your spouse wants you to drop by the store on the way home, without unlocking your phone!

I'm a Mac OS X developer and I just finished adding Notification Center support to my app. For what it's worth, here's what I found out:

• By default, notifications (the alerts) are NOT shown if the application is already frontmost (this explains why alerts refuse to show up sometimes, and you only see the notification in Notification Center). This is up to given applications to overwrite and it is not very straightforward, so do expect most applications to not show notifications when they're active. In my opinion, this is very annoying.
• The application, after sending a notification still has a lot of control over that given notification. It can remove it from your screen at any time it wants, regardless of the settings you have. Basically, if Safari, for some reason was set to remove notifications after X seconds, it will do that, your settings do not matter.

I have no idea if this is a bug in Safari or Notification Center, or both. However, I am fairly confident that there is nothing you can do about the second issue (there might be a hidden defaults setting for the first one, but I haven't heard of it). I did check the WebKit API Specification, to make sure it's not something that the individual websites are doing, and you can rule out websites from your suspect's list, because they do not have any control over the notifications.

I guess we'll have to wait and see if Apple will address this in its next bug fix release (10.9). However, by the looks of it, not showing notifications when the app is frontmost is a "feature", so don't get your hopes up.