Thanks Rolf.
Unfortunately I've hit a dead end. I've uploaded the full package as in point 1, and also confirmed allow_url_fopen is enabled by using PHP phpinfo() as in point 2.
The problem here is that the Printing Module uses a command like this to get its data for inclusion (not real, just an example):
file_get_contents("
http://www.myhost.co.uk/index.php?option=print")
The trouble is, this command makes the server connect to itself, and this results in the server rejecting the connection. I have no idea why my webhost behaves like this, but they have said it's for security reasons and they will not change it.
Oh well, at least the Printing Module is the only thing affected!
I suppose the Printing Module could be coded in another way to avoid the server connecting to itself, but the vast majority of users are probably unaffected so there's probably no point in doing it.