The LM358 is often used as a buffer or amplifier. It can be used to protect the RX or it can be used to raise a voltage level up to a usable level. Without seeing the complete LM358 I would be guessing it's purpose.
Your data that is in the PDF looks very helpful. What I don't see or have not figured it out is: what is the baud rate, parity, stop bit, and start bit counts. Serial communication has parameters that must be correct for one device to understand the other.