When I disconnect the JTAG debugger and set irq13 to 0 to let the processor to boot from the LPC dongle flash, it does not boot. Do I need to perform other operations to let it boot autonomously?
No, it should work without changes.
But how do you provide power to the dongle?
Note that the dongle uses an Altera FPGA with the logic definition stored in an external flash memory, this flash memory is read by the FPGA on power up. There's also a power-up delay for the FPGA.
In total the dongle needs a few 100 ms to boot. The Geode needs to read from the memory before that time, so if you don't power the Artec dongle externally (e.g. USB cable) you will not be able to boot from it.
My Artec dongle is powered externally via USB cable so, when processor tries to fetch boot code, FPGA should be already up and running. However I noted that R26 on dongle in not mounted thus RESET coming from processor board is not connected to the reset monitor. Is that ok?