Don't worry about the questions, it's the best way to learn stuff:
1. Is this rod (5cm) likely to be too small to get a reasonable signal? I'll probably go out tomorrow and buy a 12cm, 300uH ferrite rod aerial from maplin.
Given your distance from the transmitter a 5cm ferrite rod is VERY small. I use a 30cm ferrite rod and I am only 31Km from the transmitter.
2. What is the best kind of amplifier to use to amplify the signal? My guess would be that most op-amps would struggle around 60kHz, so it would be normal to use some kind of transistor based op-amp.
A lot of modern op-amps such as the 5532 have a gain bandwidth figure of about 10 MHz so 60 KHz is not a problem. For my DCF receiver I use the following circuit
http://www.techlib.com/electronics/antennas.html but with lots of modifications. There is no Hi/Low switch and no varactor tuning, I just use the ferrite rod plus a suitable fixed tuning capacitor. Also the 1k resistor across pins 2 and 7 is replaced with a 77.5 KHz crystal to make a VERY narrow bandpass. Finally I use both output pins of the chip and run a balanced cable to the main electronics box.
3. I guess generally what would be the next steps to detecting this signal?
The amplifier above will give you a reasonable level of signal down the cable to overcome losses. The next step is decoding MSF and there you're on your own because I don't know how to do this step.