ESP8266 + Arduino Uno: Can't send AT Commands
Tx and Rx should be crossed; meaning connect Arduino Tx to ESP Rx and vice versa. 

Your second (and first) attempts are correct. Use 1uF capacitor between Vcc and Gnd. This will provide a more
robust connection since the ESP can draw too much current which might cause it to reset. 

When using external power supply, hook together the Arduino Gnd to the PS Gnd.

If you get "Programmer is not responding" - disconnect and reconnect the Arduino USB. Also, you shouldn't get this at all since you are not trying to program the Arduino, just use its FTDI as proxy to the ESP. Any serial terminal application can do it (you don't have to use the built-in terminal in Arduino IDE).