Is there a way to hide the notifications that a given application shows in the Ongoing Notification bar?
As of Android 2.0, the system requires that apps that run constantly in the background, and that require a certain level of priority so that they don't get killed offhand (like music apps), put an Ongoing Notification. This was to prevent abuse of requesting this level of priority without letting the user know that it's occurring.
You can read about this here: http://developer.android.com/reference/android/app/Service.html#startForeground(int, android.app.Notification)
Try Smartbar. I used to use it and it allows you to put shortcuts in the notification bar as well as hide icons in the notification bar. I can't remember off the top of my head if it hides the icons put there by other apps but I think it does. Sorry for the cheap answer, I just can't remember and I don't have time right now to download it and try it out to be sure.
As others have said, you can't remove an ongoing notification from the notification bar. It's there to alert the user that the app is constantly running, and without it the app could potentially be killed at any time by the OS.
That said, if you are running Jelly Bean (or newer), there is the option of disabling all notifications from an app. The setting is per-app and can be found in Settings > Apps > (select the app) > uncheck the "Show notifications" box. The one caveat to this method is that the app will no longer show any notifications in the notification bar, but if that's acceptable to you than I've found this method to be an easy way to stop my notification drawer from filling up with unnecessary notifications.
I would talk to the developers of apps and ask them is they plan to include the option to remove the app from the notification bar. They may oblige such a request. You could also leave a comment for the app and add that as a side note.
CM team and other ROM developers do create flashable updates to their ROMs so that icons and other items can be removed from the notification bar, but they usually do not do this for specific applications. You could check out information on this in the forum of the ROM you are using.