My question is this: is there a way to send simple messages from "computer B" to the Arduino software on "computer A"?
The Arduino software runs on the Arduino, not on computer A.
An example would be that I could send the message "LED #1" to computer A
Not via the "Arduino software".
and computer A would send that through the Serial monitor software
No, it won't. You CAN write an application that talks to the Arduino via the serial port, and gets data from some other computer.
Also, try to keep this as simple as possible,
What you are proposing is not at all simple.