regards,
Raitis

You try to use SoftSerial on HardwareSerial pins. I suppose this is your problem, so try change the mode to REMOTEXY_MODE__ESP8266_HARDSERIAL_CLOUD, set REMOTEXY_SERIAL to Serial1 and remove REMOTEXY_SERIAL_RX and REMOTEXY_SERIAL_TX.

// определение режима соединения и подключение библиотеки RemoteXY
#define REMOTEXY_MODE__ESP8266_HARDSERIAL_CLOUD
#include <RemoteXY.h>
...
// настройки соединения  
#define REMOTEXY_SERIAL Serial1
#define REMOTEXY_SERIAL_SPEED 9600
...

You can change these options in the Editor.

So for 1.  I use WiFiEsp library examples WIFIClient and WIFIserver. And there I faced with limitation for MEGA, it should be connected via communication pins like Serial1 pins 19RX an 18TX. And I am willing to use same ESP module for sending the data to RemoteXY cloud.
  But in the RemoteXY sketch I can't set 19RX an 18TX, it dos not work with them.
   If there is possible to do it some hove, or what solution you could recommend?

Here is beginning of the RemoteXY code what I use:
      // определение режима соединения и подключение библиотеки RemoteXY 
      #define REMOTEXY_MODE__ESP8266_SOFTSERIAL_CLOUD
      #include <SoftwareSerial.h>
      #include <RemoteXY.h>

     //R: DHT:
      #include <SimpleDHT.h>

       // настройки соединения 
     #define REMOTEXY_SERIAL_RX 51
     #define REMOTEXY_SERIAL_TX 53
     #define REMOTEXY_SERIAL_SPEED 9600
     #define REMOTEXY_WIFI_SSID "xxxx"
     #define REMOTEXY_WIFI_PASSWORD "xxxx"
    #define REMOTEXY_CLOUD_SERVER "cloud.remotexy.com"
     #define REMOTEXY_CLOUD_PORT 6376
      #define REMOTEXY_CLOUD_TOKEN "xxxxx"

Thank you in advance for your help.