Xbox problem

G'day W0lf! You are correct in your conclusion that the HDD is the issue here. Thank you for trying all the steps in our support pages!

Going forward, the best course of action will be this:

• If you're still able to access files on your HDD, try to move your save files onto a USB flash drive or Memory Unit.
  
• Register your console and order a replacement HDD at our repair site.
  
• If you don't want to order a replacement there, I recommend you stick to trusted retailers from which to buy your new HDD, to make sure it is an official Xbox product. 3rd party HDDs will not support backwards compatibility with Original Xbox games, and
  we cannot guarantee the quality of 3rd party devices.
• Once the HDD is replaced, re-download your content. There's no need for a license transfer here as you'll be on the same console!
  

If you have any further questions, please let us know. =)

You could try to contact support and if you still have warranty you can explain the situation and if it is deemed necessary they can send it in for repair if they find anything wrong with it they will fix it if not they will send it back, just time lost.
Taking from your post I would go with the hard drive issue, make sure it is properly seated and never move the console while it is operating.

Have you had the E68 message again? If your console is having a hard time reading the discs it may need to be looked at due to the disc reader not working.

My xbox has been reading game discs fine since removing the HDD, however while the drive was installed it would either say the game disc is damaged/unreadable (when it was unscratched/new) or that the download failed (despite it being on-disc play). I've
been able to load games up and play them for hours, save them to the mem unit without issue (with the HDD removed). So does that sound more like a bad/corrupt HDD rather than a faulty 360 (since the former seems more likely at this point after my investigation)?

I cannot access my xbox live account..

It just says Windows Live ID invalid.