When I hold down the EN button and plug the USB into my laptop, it flashes on two seconds and off two seconds which is the correct flashing for AT command mode. (As it should). However when I send the code on post #21 to Arduino, and open the serial monitor to Both NL & CR and 9600 baud. I then attempt to send "AT" without quotes, or any other AT command it does not say OK.
When I don't hold down the EN button and its not connected, it constantly flashes on/off in under a second (As it should).
Your AT mode and unconnected data mode flash patterns appear normal, however it's clear that you are not seeing a response to AT commands.
Is it still true that when you pair and connect a phone to the module you still see the abnormal flash pattern and the delayed transmission?
What brand of phone are you using and what is the Android os level?