WRT the serial level shifter, the goal is to take the serial data coming into the Uno, capture the variable values, and pass those along to the neopixels, which are now connected to the Uno instead of the ESP, with the required 5V IO level.
![Esp8266 serial communication to arduino Esp8266 serial communication to arduino](/uploads/1/2/5/4/125464630/633819109.png)
Since you are Arduino Uno, it has only one Serial. This Serial can be used to transmit data in USART mode to another Serial Device as well as see any Debug data in the Serial terminal. However, both the functionality can not be used simultaneously. The Tx Rx pair of Arduino can be connected to one pair of other device i.e. Either to the computer via the cable or to the Tx Rx of ESP8266.
But if you really need to plug in the Uno to the computer and send data to ESP8266 simultaneously, then you have to use library to convert any digital pin to Tx Rx pin, and connect them to ESP8266 for transmitting data.