I also had ordered a Nano last year from Amazon under the same product description, but from Computing Alternatives. However, what I received was a "Nano V3.0 (Arduino-Compatible)". Yes it says that on the board. But it is working fine for me.
From what you are saying it seems your FTDI chip is working and the ICSP interface is working, but the serial pins to the mega328P is possibly not working.
To confirm the FTDI serial loopback working:
- On the Nano you jumpered the RST and GND pins.
- jumpered the Rx0 and Tx1 pins.
- Connected up the USB to the Nano and it showed up on COM4.
- Selected Serial Port COM4 in the IDE.
- Opened up the Serial Monitor.
Whatever you typed in the entry window is echoed back in the window below. This should confirm the FTDI chip is working correctly. Also for this test you should be able to choose any baud rate and it should work. The standard bootloader for the Nano uses 57600 baud. So you may want to try that speed for testing the loopback.