What is the German / European equivalent of FAA Order 7110.65? FAA Order 7110.65 prescribes air traffic control procedures and phraseology for use by personnel providing ATC services in the USA.

Which documents regulate these aspects in Germany, or, more generally, in Europe?

Please provide also the valid versions of these documents as of the 1st of April 2020.
The „handbook“ for ATCOs in Germany (!) is the MO-ATS (Manual of Operations for Air Traffic Services) which is an internal DFS (Deutsche Flugsicherung - the German ANSP) document and therefore not available to public.

However it is based on the German AIP and ICAO DOC 4444 which are both available to public. For the international legislation check Milo Morkus’ answer.

In the German AIP phraseology examples are found in AIP ED GEN 3.4 Communication Services