Are all ATC's around the world, especially those at big airports required to use English when talking to pilots, and vice versa?

English is the official aviation language. However, it is still allowed for ATC to talk to local traffic in the local language, all international carriers must speak English, and even a local International carrier will use English.

 

These requirements arose from strong recommendations after the mid air collision over Zagreb.

 

http://youtu.be/uaQ7QHPWnmQ

Are all ATC's around the world, especially those at big airports required to use English when talking to pilots, and vice versa?

The minimum requirement for both ATC and any pilot flying within controlled airspace is ICAO level 4 equivalent to Cambridge 1st Certificate. Airlines such as Alitalia for example require their pilots to attain ICAO level 6 which is equivalent to Cambridge Proficiency (the highest level a non English speaker can get).

Any airport that has scheduled flights of any description must have English speaking ATC. En route controllers must also be proficient in English.