You do not need to have an SD card installed. And as you figured out, you need to use "Compile to Target" instead of "Compile Project to Target".
You definitely need "compile to target" and "store program in flash" set, unless you are going to compile to a .bin file and upload it to the device with RFU (Rabbit Field Utility).
You can keep your printf() calls in your program -- the application will skip that output when you aren't running a debug build. It's also possible to redirect stdout to a serial port on production builds. See Samples/STDIO_SERIAL.C. If you redirect to serial port A, you can use the "DIAG" connector of the programming cable to monitor the output.
I would try running Samples/RCM4300/FLASHLED1.C or FLASHLED2.C. Compile to the board and make sure it works correct in the debugger. Then disconnect the programming cable and reset the board to see if that works.
Once you have those samples working, you know you've configured the compiler correctly to install a program to the core module, and the board is properly powered/grounded.
You can make use of printf() and redirected STDOUT to debug attempts to run your networking code outside of the debugger.