How can I turn off the annoying Notification sounds but still leave phone ringer on?
Not sure what version of android you have, or if this is different between versions, but in the settings there is the option to Not Use Call volume for notifications. If you uncheck that, you get a volume slider for notifications.
Audio Manager lets you control this, it's what I use.
I have Android 5.1 - and after googling/browsing this question, found there is a 'None' setting for default notification sound.
Settings, Sound & notification, Settings icon (Cog symbol right of General Ring), Default notification sound, scroll up to the first, select 'None', OK - job done.