ESP8266 problems

Depends on the type of ESP9266 board you have… there are a few :wink:

Yes… well sorta… you should be using Software Serial with an UNO, so make a few changes in the sketch and use digital pins 2 (RX) and 3 (TX) instead.

// Hardware Serial on Mega, Leonardo, Micro...
//#define EspSerial Serial1  // Comment OUT this part

// or Software Serial on Uno, Nano...
#include <SoftwareSerial.h>  // Use these thse parts
SoftwareSerial EspSerial(2, 3); // RX, TX

Again, depending on the type of board you have.

If you have a full Dev board with built in USB, then don’t bother with the Arduino… follow this tutorial instead

http://help.blynk.cc/how-to-connect-different-hardware-with-blynk/arduino/esp8266-standalone

If you have something entirely different, let us know what it is.

Shared GND is necessary whenever you have multiple PSUs (like Arduino on USB and ESP8266 via your regulator.

But if you built your own 3.3v regulator circuit, then the cap is for smoothing (optional, but recommended) but the resistors are not required… as they are used to drop from a 5v to 3.3v supply as per the shown diagram that shared the single 5v PSU for both Arduino and ESP8266.