Does ICAO require (or suggest) set of technical statuses of radar and its subsystems?
These statuses are not defined by nor required explicitly by ICAO. 

The details of how a radar (or other surveillance) system communicates with downstream ATC systems are defined in a technical specification documents, typically produced by EUROCAE and/or RTCA. 

The ASTERIX specifications are maintained by EUROCONTROL and only give the format in which surveillance ground systems communicate amongst each other. The specifications were originally intended to facilitate surveillance data exchange between European countries, but they have now been adopted by many countries outside Europe. 

Category 002 (old) and 034 define the status messages monoradar systems use to update downstream systems (radar trackers, displays) on the radar's status.