I have now tested, with having RX at TX, TX at RX. And then taken out CH_PD once, then I get the "Failed to assign IP address to ESP8266.", error. I tried again, using all the connections as in the documentation, but taking out the GPIO0 connection. This time I also got "Failed to assign IP address to ESP8266.".
When I have all connections the same as in the documentation, I get the COM error. Could it be that the COM changes, when the arduino is connected to the external wifi module?
Also, when I did the arduino IDE example in the esp6288 documentation I got a connection error;
connecting........_____....._____....._____....._____....._____....._____.....____Traceback (most recent call last):
and then:
esptool.FatalError: Failed to connect to ESP8266: Timed out waiting for packet header