LCD Backlight Control Circuit
From the datasheet, the LED is not a logic control input, it is direct power supply pins to the internal LED(s). So, getting rid of R3 is not an issue.

R3 probably does not serve any purpose in this circuit, anyways. When the transistor is cut-off, being a non-linear device by itself, the LED itself is turned off. Hence, R3 does not serve any purpose. When the transistor is driven to saturation, the LED is turned on, and R3 is just in parallel with the LED. serving no purpose.

On the pull-up question, you generally need pull-up usually only on logic pins or pins that have a defined behaviour that is unwanted when floating. The selection of pull-up value depends upon the leakage current on the pin being pulled-up. It could be obtained from the manufacturer's datasheet.