Arduino WEMOS and RAMPS
After having looked into them, these boards certainly seem to be rather interesting.

There appear to be two varieties of this board, as I have seen photos with either RobotDyn:



rear (from Instructables)



or WeMos logos on them. 



They may, or may not, be the same. There is also this one, with just a generic "WiFi" logo (from the eBay page below):



Memory Discrepancies

However, there is certainly some discrepancy relating to the specification and amount of memory (unless there are a number of variants - which would certainly be possible):


4&nbsp;MB of Flash is in the ESP8266-12, as Dan states, see ESP8266 SMT Module - ESP-12. But is it the ESP8266-12 in the board, or is it another ESP variant?
This WeMos product, on eBay, WeMOS Mega + WiFi R3 ATmega2560 + ESP8266 USB-TTL For Arduino Mega NodeMCU appears to agree with the 32&nbsp;Mb1 
This RobotDyn board, Контроллер RobotDyn Mega 2560 R3 + WiFi ESP8266b, flash память 8Mbit, USB-TTL CH340G, MicroUSB, has 8&nbsp;Mb apparently. 
However the same product page, but in English, upgrades the memory to 32 MB! See MEGA+WiFi R3 ATmega2560+ESP8266, flash 32MB, USB-TTL CH340G, Micro-USB


Looking at this article, Instructables - Arduino MEGA 2560 With WiFi Built-in - ESP8266, it isn't exactly the ESP8266-12:


  In today's text, we discuss an Arduino that I consider extremely special, as it has an ESP8266 embedded in its board. It doesn’t have the ESP12 soldered onto the board. Instead, it has the Espressif chip. So, on the board you have the built-in Tensilica chip with 4MB of memory, along with the ATmega2560, which is the traditional Arduino Mega.


So the ESP has 4&nbsp;MB, but where does the 32&nbsp;Mb come in to play?

Well, looking at this RobotDyn memory table (taken from the Instructables page):



The 4&nbsp;MB of the ESP appears to have been immediately contradicted and upgraded to 8&nbsp;MB! The ATmega2560 somehow has 32&nbsp;Mb (which seems a bit of a lie - as there isn't an ATmega2560 known to man that has that much memory2), and is confused further by the corresponding text:


  As we see in the table above, the ATmega has 32MB of memory, not counting the ESP memory. This is wonderful, since the traditional Mega Arduino has only 256kb of memory. 


So MB or Mb? Is there an additional flash IC external to the ATmega2560? Looking at the schematic, taken from the Russian RobotDyn page, there doesn't appear to be:



So this could be nonsense. Logic would dictate that the ATmega has 256&nbsp;kB and the ESP has 4&nbsp;MB... But I could be wrong..?

Links from Контроллер RobotDyn Mega 2560 R3 + WiFi ESP8266b, flash память 8Mbit, USB-TTL CH340G, MicroUSB:


Schematic (PDF)
Pin out (PDF) (JPG)
Mechanics (PDF) (JPG)




Reliability of communication

Regardless of the actual spec, to address your question of ESP/Arduino communication reliability:

As the ESP8266 and the ATmega2560 are on the same board one would assume that the (serial) communication between the two would be faultless. However, some issues have been reported3. 

Nevertheless, from your question, it seems that you were having issues with the ESP to the OctoPi communication, rather than ESP to Arduino, so maybe this doesn't actually answer your question.

Also, as to their compatibility with RAMPS, there is very little information available currently, and it would seem to be a case of try-it-and-see. The closest I got was this post on the thread, OnStep and RAMPS1.4, but it seems to be spam. 

However, it certainly sounds worth a gamble as it would be a nice combination of technologies. However, I would imagine that some modification of the firmware would be necessary.



Useful Notes


From post #8 of Mega + WiFi R3 ATmega2560 + ESP8266 (8 Mb mémoire)



  To upload the sketch to the board:

5, 6, 7 ON
RXD3 - TXD3   

  
  To run the sketch:

5 and 6 ON
RXD3 - TXD3




Here is an excellent manual in Russian




Footnotes

1  From WeMOS Mega + WiFi R3 ATmega2560 + ESP8266 USB-TTL For Arduino Mega NodeMCU


  WeMOS Mega + WiFi R3 ATmega2560 + ESP8266 USB-TTL For Arduino Mega NodeMCU 　
  
  Features:
  
  
  Full integration on one board: Mega R3 ATmega2560 and WiFi ESP8266 with memory 32Mb (megabyte)
  All of the modules can work together or each separately. And everyone has their own pinout headers
  The convenient solution for the development of new projects requiring Uno and WiFi 
  Via USB you can update sketches and firmware for ATmega2560 and for ESP8266. For this on board have the USB-serial converter CH340G
  Use this board is very simple
  The board has DIP-switch, to connect the modules. For example to:
  
  
  USB and ATmeg2560, 
  USB and ESP8266, 
  ATmega2560 and ESP8266
  
  
  
  Table DIP- Switch:

Connection                                                          DIP
                                           1        2        3         4       5         6        7
ATmega2560&lt;-&gt;ESP8266                      ON       ON       OFF       OFF     OFF       OFF      OFF
USB &lt;-&gt;ATmega2560                         OFF      OFF      ON        ON      OFF       OFF      OFF
USB&lt;-&gt;ESP8266 (Update firmware or sketch) OFF      OFF      OFF       OFF     ON        ON       ON
USB&lt;-&gt;ESP8266 (communication)             OFF      OFF      OFF       OFF     ON        ON       OFF
All independent                           OFF      OFF      OFF       OFF     OFF       OFF      OFF

Special solution:
USB &lt;-&gt;ATmega328&lt;-&gt; ESP8266
Connection                                                          DIP                                     SWITCH 2
                                           1        2        3         4       5         6        7

USB &lt;-&gt; ATmega2560&lt;-&gt; ESP8266             ON       ON       ON        ON      OFF       OFF      OFF         To RXD3/TXD3

  
  
  USB converter CH340G connect to RX0/TX0 of ATmega2560
  ESP8266 connect to RX3/TX3 of ATmega2560
  
  
  Size: 10.5x5.3cm/4.13x2.09inch


2 From the Atmel ATmega640/V-1280/V
-1281/V-2560/V-2561/V datasheet


  


3 From Mega + WiFi R3 ATmega2560 + ESP8266 (8 Mb mémoire), there do appear to be some issues communicating between the ATmega and the ESP ICs, with a number of users reporting problems. 

However, a couple of solutions are provided:

Solution 1

From post #2


  
  Set the DIP switch to "special" mode: (1=on, 2=on, 3=on, 4=on, 5=off, 6=off, 7=off, 8=off) see the the description of the
  product.
  Set the switch in RXD3
  Run the below sketch, change the SSID and PASSWORD values. this is just to test that Arduino and ESP8266 are working together. The code
  originated from BISSA ORBOEA in Element14.
  
  
  Code:

#include "ESP8266.h"

#define SSID        "YOURSID"
#define PASSWORD    "YOURPASSWORD"

ESP8266 wifi(Serial3,115200);

void setup() {


  Serial.begin(9600);              

  Serial.println("Restart");
  wifi.restart();
  delay(1000);

  Serial.println("FW Version:");
  Serial.println(wifi.getVersion().c_str());

    if (wifi.setOprToStation()) {
        Serial.println("OprToStation ok");    }
        else {
        Serial.println("OprToStation  err");
    }

    if (wifi.joinAP(SSID, PASSWORD)) {
        Serial.println("Join AP success");

        Serial.print("IP:");
        Serial.println( wifi.getLocalIP().c_str());      
    } else {
        Serial.println("Join AP failure");
    }
}


void loop()
{
}



Solution 2

From post #6


  First you need to write the code for each board according your
  application, but in your code you need to establish a commmincation
  between serial port of Atmega and ESP in both code, again in according
  you application. Finally, put the switches which make the intenal
  interconnection between both. 
  
  For example, I made a code for ESP to connect in an Access Point and
  put on a TCP server. When a wireless client is connected in ESP and
  consequently in the TCP Server, all data send by client I redirect to
  the serial port. As serial port of ESP is connected with serial 3 of
  the Atmega, the code in Atmega pass to theat the data from serial 3.


Solution 3

From post #11


  Follow this if you have the same to my cases:
  
  
  programm to the ESP(upload sketch) for wifi connection by configure the DIP to 5:on,6:on,7:on and other off.
  program to mega(upload sketch) in case you want to send communication to each other by configure Dip to 3:on,4:on, and other
  off.
  switch two  RX0/TX0(in my case I use Serial.print() and Serial.readString() for communicate between Esp and Mega)
  after you already upload sketch to both Modules you can test communication between them by configure DIP to: 1:on,2:on,3:on,4:on.
  and then press Reset Button to restart the boards.
  In ESP you can get the string from Mega by  if(Serial.available()) Serial.readString();  and In mega you can get data from ESP by :
  if(Serial.available())Serial.readString().  
  After you got a string so you can split or do anything whatever you want. 
  If you want send data to Firebase you must use the Firebase Library at the Esp side, in Mega it not support.