Grumpy_Mike:
You ignored that didn't you?
Can you read that schematic? Let me tell you, you can't.
The parts need dragging into a more sensible place. Then the dotted wires which are called rat's nest wires need routing so they are composed of horizontal and vertical sections. Diagonal wires are not acceptable in a schematic.
The thing about Fritzing and it is why it is hated on this forum is that to use it you require far more skill than you have when you use it, which happens when you feel you need a physical layout diagram. So you make substitutions and render it all useless. By you I mean one, not you personally.
Where is the 5V coming from?
The problem with this is the connections to the Arduino. When the output is high you feed 5V into a 3V3 system, this is very bad. When the output is low the Arduino is feeding a negitave voltage into the ESP, due to the voltage drop across the FET.
Either of these can be enough to latch up the ESP and stop it working.
The use of power switching from the top like in reply #10 will solve the negitave voltage situation. A potential divider of 510R and 1K will cut down the 5V output signal from the Arduino to the ESP to 3V3. Finally you need to make that output signal into an input before removing the power, and make it an output again once the power has been restored.
Mike, I wouldn't say that I ignored, I just did not understand how to do that. I'm not an engineer just an enthusiast who is trying to make something functional.
Regarding your question about 5v, I don't have any 5v here, if you see the breadboard you will see 3.7v battery, my actual input is based on this:

The reason why I don't have this in the breadbard/schematic from fritzing is because I didn't find the components to do it.
But I can ensure you that ESP is receiving 3v only and not 5
I know it is not tolerant and you can burn the whole thing if you insist on that.
I reviewed your schematic from #10 reply, looks very good, but there is something I need to ask considering my limited knowledge on this: Is there an easier way to make it work even if is not the best way ? If not then I'll try to implement it. I ordered already the 2n3904 that should get home in few days.
Many thanks for your patience !