Can advance tickets that were purchased online be picked up at any ticket machine at any station? During checkout, you are prompted to pick a station where you would like to pick them up, but are you then restricted to only that location? Thanks!

No, you can use any FastTicket machine on any line, but some stations don't have any, or only have a a ticket machine that doesn't allow printing of prepaid tickets.

I've never tried to collect tickets from a station other than the one I specified online. I'd assumed that, because you are asked to choose a station, then the tickets need to be collected from a machine at that station.
I have booked tickets in this way in France, Italy and Sweden, but their systems do not ask you to choose a station, and the tickets can be collected from a machine at any station in the country

They only ask you to specify the station because not all stations, which may include the starting point for your journey, have FastTicket machines. But you can in fact collect your ticket from any compatible machines. You need your credit card and your reservation code.

Thanks for your help, Alec and GeoffHamer!

Originally, only the bookings for that specific station were donwloaed for collection into that station's machines. So you could only collect from the station you specified.

They've now changed it, and (far more logically) the machines interrogate the central database and produce your tickets whatever your original choice of collection sttaion.

But not every station has a machine, the smallest won't have.