I’m living in university accommodation and I can’t update any program by connecting to its server, I have to locate and download files manually.
I’ve just installed RA3 and it can’t connect to the server and verify the game (as it thinks that my computer is offline) and so I can’t play it. Please help 
Sounds like your the admins for your university have blocked the ports the game uses. Can any of your mates using the same connection play?
I’m almost certain that your universiy has a hardware firewall in use. You could always ask a Network Admin there, if possible.
Would this work; could I remove my computer’s hard drive, put it inside another computer that is outside the university network, verify the game there and then put the hard drive back inside my own computer?
Only if the other computer has almost exactly the same motherboard chipset.
Best to take your whole computer over to a different network 
Even if he gets verified he still might have trouble playing online.