Thanks for the helpful response. The pc has been with the manufacturer for a few days, and they carried out various tests including memory tests, but have been unable to replicate the crashes, so have no diagnostic information to go on. They suggested running the pc disconnected from the network for a few days, to see what happened. So far, so good. Then I shall connect to the network, and keep a record of what happens. Yesterday I had 2 ancient XP pcs on the network running all day, and a laptop with Windows 7 64-bit. I transferred several Gigs of files between one of the pcs and the laptop, and there were no problems with the network. I am beginning to suspect the network card in the new pc, or its driver, and am thinking of installing Driver Genius or similar to see if updating drivers may make a difference. I guess I will need to be patient.
The problem with Firefox seems to have been that the file on the server was geuinely corrupt, as I managed to install it a couple of days ago from a different server.