Spotlight not connecting to internet
This seems like the sort if thing I would update to the latest OS version (you appear to be 10.11 - the initial and possibly most buggy release possible) and then retest with 10.11.6.

If you still have problems - make a new user account, log out of your current account and then see if a new account with default settings can search.

Unless you are on a corporate network that blocks some outbound connections - this likely is an issue with the settings file or an odd bug.