The purpose of the serial cable is to let you see what the board is doing when booting.
If you're sure the cable is correctly made, then it sounds like you're telling PuTTY to listen on the wrong port.
Have a look at what ports you have in device manager when the USB cable is disconnected.
If all of that looks right, then try swapping pins 2 & 3 to see if you get anything then.
Make sure pin 5 is ground; you'll want the same ground on both computer and board.
And yes, you can boot the board with the 3.5mm plug installed.
I don't believe the board has any persistent storage available for sketches w/o an SD card mounted;
i.e. it won't survive a reboot.