Facebook Intersect Search does not work

1. No. It's integrated to Facebook search algorithm. Public don't have access to it.

   This has been made possible recently (near the end of 2012). Here's how you can delete your search history:

   • Go to your profile and click on "Activity Log".

   

   • On the left sidebar, you will see options like Photos etc.

   

   • Expand them if needed by clicking on "MORE" until you find one that says "Search". Click on it.

   

   • You will see all your Search history. Click on the link that says "Clear History" on top of it.

   

   • Click "Clear Searches" on the confirmation box that appears as the last step.

   

2. No. That's not public. Some apps claim to show you who visited your profile. But they are just scam. It's not possible. Facebook doesn't provide such information.

In AdBlock Plus or a similar browser extension that can use ABP filters, simply add a Facebook Annoyances blocking list at facebook.adblockplus.me to ABP.

Its 'Block news feed annoyances' list blocks "Page X is posting about Page Y" content in your news feed. I recommend the 'Block all Facebook annoyances' list which blocks that and more Facebook annoyances.

If you just want to block "Page X is posting about Page Y" content, then just add the following custom filter to ABP (this filter is also from the Facebook Annoyances blocking list):
facebook.com##.storyContent > .storyUnconnectedPost