Trying to flash ESP8266 with Arduino UNO
You do not need the level shifter. The input pins of the ESP8266 contain over-voltage (though not over-current) protection. All you need to do is limit the current that can flow through the over-voltage protection circuit, and all that needs is a series resistor in the region of 100Ω or so.

And the Arduino already has that protection between the ATMega16U2 and the TX/RX pins anyway in the form of 100Ω resistors that it uses to prevent short circuits between the two MCUs on the board, so you don't need to add it in this instance (you do if you want to communicate between the ATMega328P and the ESP8266, but only between the Arduino's TX and the ESP8266's RX pins).

When it comes to actively running the ESP8266 it can be beneficial to power it from a more powerful 3.3V regulator than the 150mA one on the Arduino since when transmitting the ESP8266 wants more than that, but for simple flashing of it the Arduino's 150mA regulator should do the job fine.

One thing to note though: I don't like the look of the soldering on your ESP8266 pins. I would be inclined to re-solder those joints. Add a bit of flux to help the solder flow properly.