If you don't want to just turn off Home Sharing on your computer, then two things come to mind.
If you changed the Apple ID somewhere else than the iPod settings - then that would explain things. (I've accidentally cleared the store account when I meant to change this upon going to a friend's house)
I just tested on my iPad and deleting the home share as depicted above immediately disconnected from my Home Share - I didn't need to kill iTunes on the iPad or do any re-starting - the effect was immediate as soon as I exited the settings tab.
Assuming you got the place correct, then your iPad iTunes database is a little corrupt. Sometimes syncing the iPad with a different iTunes library (music only) will delete your music and settings and allow things to work again. It's not easy to poke in the files directly on the iPad - and you may have to restore the device and set it up as new if getting a new set of iTunes data doesn't resolve it.
Lastly - you could check troubleshooting steps with Apple - the express lane at the bottom is good if the steps listed don't resolve it for you. It's highly unlikely to be something on your account with them or the computer, but they will know the oddball causes if the obvious next steps don't work.
iDevices are designed to be linked to an iTunes library where they get their content. If you've never trusted iTunes before and been manually copying items onto your iDevice then you're just making it hard on yourself.
Synchronizing your files and upgrading the iOS are separate steps. I would recommend you get the iPad linked to your iTunes library properly and synchronization figured out before trying to upgrade lest you lose your iPad content in the process.
First, make sure your iTunes and your iPad are registered to the same AppleID you used to download your apps and other iTunes store items. (This may already be the case if it's telling you about purchases, but check.)
Next, download your past purchases to your Mac.
For other items like your own music and video files, the synchronization is one-way. If they're not in your iTunes then they won't sync back from your iPad.
Try using manual mode for those items if you're happy leaving well enough alone with keeping those items on your iPad.
Otherwise, you'll have get these from your original source (ideally) or use a third-party utility to extract the files from the device. It's been a while since I've had to do that myself, but as I recall it was a bit messy since the files lost their original filenames when they were stored in the device's filesystem/database. At least I think you can get them out somehow, but yes, it might be buggy.
The key is to have everything in iTunes before allowing the iOS upgrade to remove anything from the iPad. In my own experience, everything was right back where it was after the iOS upgrade, even though I had some similar warnings, but if you don't already have the items in iTunes then it might be a different story.
Best Answer
The best way to do this is to connect the iPad to iTunes and do a 'sync' of everything, including its applications. This saves the iPad's applications to your local computer.
After doing the sync, you should then do a backup. This will save its settings.
The reason why it is important to do the sync is because there is the possibility that some applications that you had on the iPad are not available on the App Store any more. Imagine, for example, that you had Flappy Bird installed on there :)
After doing the restore, you should then sync the iPad with iTunes again - this will put all the content that was on the iPad before you restored it, back on the iPad.
If you do not do this, the iPad will only be able to get any applications that are still available on the App Store now, and any apps that are no longer available, will not be on the iPad any more.
EDIT: Apps coming back as being "not compatible", may be apps that used to be compatible when they were created ages ago, but have not been compatible with the original iPad for a few years now, and there is no longer a compatible version on the Store.