ESP32 not seen by a network analyzer application in server mode
In Client mode the ESP is using the WIFI capability. But why it can
  not be seen by others?


The app is looking for access points. (if it is looking for SSID) And if you do not have the ESP set as an access point it will not be seen. 


  why is the ESP32 Lolin getting hot only when it is in AP mode?


Because the radio is on all the time. It has to be available for a connection. When it is not an access point, the ESP goes into a sleep mode when it does not have any need to communicate.