As for the main problem - I, er, um, er, hadn't connected the DTR pin at all

Now it's plugged in it works so much better. Sorry.
That looks correct to me.3.3v -- (Not connected)
GND -- GND
VBUS -- VCC
TXD -- RXI
RXD -- TXO
DTR -- DTR
Again this seems like the driver is installed correctly."CDM v2.10.00 WHQL Certified.zip" from the FTDI website and I now see "USB Serial Port (COM6)" in Device Manager.
That's a good sign as it mean that the pro mini is alive and running code.On the Pro Mini, there's an LED near the end connector that glows brightly and another LED at the opposite end of the board that flashes continuously.
This means the Arduino IDE has correctly managed to reset the Pro Mini, which then reboots into the bootloader, which in turn should accept your new sketch.(4) The TX LED on the adapter flashes a few times (not very brightly);
This is a generic error message and basically means the the IDE is not getting any kind of response back from the pro mini. The main causes of this is either a bad connection or the Tx and Rx pins being connected in the wrong order. However it looks like you have that bit set up right.(6) After several more seconds the progress bar in the IDE goes to 100% complete and the following error is displayed:
avrdude: stk500_getsync(): not in sync: resp=0x00