Why shouldn't i power ESP8266 WiFi module directly from arduino nano 3.3V rail?
The 3.3V rail of the Arduino is not meant to provide much current. The onboard regulator is a simple linear voltage regulator, which will dissipate the extra power as heat. As you draw more current through it, it will get hotter and finally get fried, as it cannot dissipate the created heat. The ESP chip needs multiple hundreds of mA, especially, when it is connecting to a wifi network. The onboard voltage regulator of the Arduino cannot handle this much. So you cannot power it over the Arduino.
I don't really know, what you mean with "using resistors".
Could i use something like a Mean Well Switching Power Supply, 5V 10A to power the Arduino Nano [...] connect ESP8266 directly to arduino without fear of breaking something?
Yes, you can power the Arduino with this power supply, but you should power the ESP through the Arduino. Not through the 3.3V, nor through the 5V rail (which will fry the protection diode, I think). Instead you should power the ESP directly from the power supply, in parallel to the Arduino. Often ESP modules have their own voltage regulator, that regulates the voltage from 5V to 3.3V. If that is the case for you, you should use this input of the ESP module with the power supply. If you don't have a 5V capable regulated input on your ESP module, you will have to use a dedicated voltage regulator for that. In that case be sure, that the voltage regulator can provide enough power for the ESP (with some headroom on top).
Could Arduino nano be powered through USB? And if yes, would it be then safe to connect ESP module to arduinos' 3.3V rail?
Yes, the Arduino Nano can be powered over USB. But again you must be sure to not draw too much current from the Arduino. Between USB and the 5V rail is a diode, that will be destroyed, if you draw too much current through it. And the standard USB port can only provide up to 500mA. And regarding the 3.3V rail: All the explanations above about the 3.3V rail apply here again, since the USB voltage (5V) is regulated down to 3.3V by the linear voltage regulator. So: No, you still cannot power the ESP through the Arduino. It's not about the power supply here, but about the electronic parts inbetween.