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And the strangest thing is the searches they do:
The urls of my site look like this:
What could this be? What are they doing?
New Search Box Under Sitelinks [webmasterworld.com]
Those new search boxes use the site: operator. They may be testing your domain to see if it should be awarded the site search box.
1. In this case the IPs are NOT googlebot - they are a user from a different IP at Google.
2. A spider requests a URL, but these look like referers from a Google SERP after a sirte: operator search.