Does ICAO require (or suggest) set of technical statuses of radar and its subsystems? Does ICAO require (or suggest) set of technical statuses of radar and its subsystems? In other words is it required that radar status can be set to Normal, Warning and Error, or Idle, Alarm, Unknown also possible? There are 002 and 034 categories of ASTERIX which describe messages about radar status, but I haven't found out answer on my question.
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.