SWO is basically an async serial output (from MCU -> whatever you want). I find it pretty handy for development, but it's indeed not required, it's just a plus.
Your wiring looks correct, I checked the official Cortex debug connector for the pin-out.
The series resistors are probably already part of your debug probe, so you don't need to add them to your cable. (And if not, it's not a huge deal here. They are mainly to calm down EMI.)