Best practice would be to run a virus scan on the DVD to check for any infected files. Then you'll know. However I think if its been burnt in the manner of a write once dvd-r - i.e. a collection of files put together and burnt in one session, then it's unlikely to be infected with a virus if used on an infected computer. If it,s been prepared for packet writing - generally this means formatting a Dvd-rw so that you can use it in the same manner as any other disk drive, then a virus could infect that as it would be able in theory to overwrite files on the disk.
Really interesting question. I would like to see a hardware solution to problems like that. I suspect that the answer to your question is that if the computer can write to the dvd then some viruses can, too. But if you had a manual switch to enable the drive to write, you could leave that disabled most of the time - especially at times like this. The minor inconvenience would be the lesser evil. Good luck:-)
The problem is no antivirus could find anything on the infected computer. so a clean scan result for the DVD mightn't necessarily mean it's clean.
I knew I had something because I couldn't go on bleepingcomputer or antivirus software sites and my computer was slow. I tried literally every scanner there is but couldn't find anything at all. I tried all the rootkit scanners too and everything came up clean. That's why I decided I would just have to dump it and buy a new clean PC.
So you've never actually diagnosed what was causing the problem then?
It is a multi session rewritable DVD so I suppose from what you're saying it isn't worth the risk?
If it's a disc that has been formatted as a live file system that you can use like a USB flash drive, then yes potentially a virus which spread by file infection (and not all malware spreads like that) could spread to the DVD, as it would possibly be seen as just another disk drive. The difference between a live file system and mastered dvd's is explained here - http://windows.microsoft.com/en-gb/windows/format-cd-dvd#1TC=windows-7
I suppose documents and photographs could be printed from a new cheap computer and printer which could be thrown away afterwards (computer no.4) and then the pages couild be scanned to the clean one (computerno.3)?
Not without losing quality. Photograph files would be most unlikely to be used as a way of installing malware, and while documents could be infected with malicious macro's, these should be detected by competent AV software. Malware is far more likely to be in compressed archives (e.g. zip, rar, 7z files) or executable files (.exe, .pif, .scr and so on) than in photos or documents. And no offence, buying a new computer only to throw out seems more than a bit paranoid and illogical. I can't think of any reason why you would you need to "throw out" any computer because of malware! Malware operates at the software level and does not permanently infect computer hardware. In a lot of cases it can be removed without destroying the operating system installation. In cases where it is so severe and so entrenched into the OS that it cannot be realistically cleaned (e.g. some rootkits) a re-format and clean install will be the only 100% certain way to get rid of it. See http://www.bleepingcomputer.com/forums/t/287710/how-malware-spreads-how-did-i-get-infected/ for some more general background info on malware.
Maybe the music could be recovered by line in recording using an mp3 player? Would that be safe?
Again not without losing quality. Straightforward music files (.mp3, .aac etc.) as far as I know cannot be used to carry out malware infections. A common "trick" is to disguise a file as another type for example music.mp3.exe could arrive by some method and actually carry malware, while an unsuspecting user could be fooled into thinking it was a music file.
Hypothetically , although very time consuming, I suppose you could transfer video files to VHS through your infected computer to your tv, and then back to digital from your tv to your clean computer?
Again you will lose a lot of quality and again it's likely to be a waste of time, because straight video files (.avi, .mp4, .mkv, .wmv etc.) cannot be used to install malware, as far as I know. A far more likely scenario for video's to spread malware would be for a dubious video streaming site to insist you install a codec to watch a video, which could be used to install malware.
I'm just thinking out loud here.