because I can’t seem to run Arduino, processing, and the Arduino serial monitor at the same time.
There is no "seem to" about it. You can't run all three at once, because the serial port only has two ends.
Processing can, however, read serial data, and print to its monitor area...
For some reason, I think the Arduino, processing , and the Arduino serial monitor are all competing for tty.usbmodemfa141, instead of sharing it. How do I get them to all use 141 without conflicting?
They are. You can't. See above, though.