Can you power the ESP8266E-12 with any of the 3.3V pins? If I want to operate the NodeMCU on the 3.3V input with 2.5V to 3.6V; can I use any of the 3.3V pins to do so?
On the nodeMCU board all the 3.3v pins have continuity. That means they all do the same thing. And yes, you can apply ~3.3v to them. I have trouble with voltages under ~2.8 though, 2.5v is likely not going to work.
Use a buck+boost to keep it 3.3v if needed. For cheap dc/dc converters, I strongly prefer ones with XL semi chips; clean enough to please the ESPs.