Regarding the Gmail Notes, I'm messing around with my iPhone right now, and here is what I am seeing. If you go to Settings > Mail, Contacts, Calendars > then choose your Gmail account, you should see these options on what to sync: Mail, Calendars, and Notes. If you choose OFF for Notes, it will stop sending your notes to Gmail.
However, I've noticed that when I went to the Notes app > Accounts while my iPhone and Gmail Notes were connected, all my new notes were only being written to Gmail. Meaning that when I disconnected the two, all the notes I created when I had Notes enabled on my Gmail account disappeared from my phone. I'm not seeing a way to copy them over, so if you want the Gmail connection gone, it looks like you will need to rewrite them.
However, Gmail connection or not, the notes are still private, unless someone else has access to your email.
As for LinkedIn, you are referring to the Calendar Sync in the LinkedIn app, right? It doesn't share it with anyone else, it's private to you. But if you don't want the Calendar stuff to be connected in the LinkedIn app anymore, go to the LinkedIn app, press "in" in the top-left corner, then "You", then the gear in the top-right, Add Calendar, and switch the ON to OFF.