Well, if it does check out okay (in the overall operation) ... then you can restore your data. That personal data shouldn't afftect the system operation itself. I say it "shouldn't" and that's the idea ... but ... there is an outside chance that there is something going on there that we might not understand. I'm just saying that by restoring it totally to factory default, you get to see how it operates completely and totally as Apple intended, without any of your stuff. THEN ... restore your stuff and see what happens. It should operate the same way, as in factory default mode ... hopefully. That's the theory, anyway ... :-) ...
Now, if it doesn't (and I knonw this is a lot of trouble doing this way, but you're finding out something when you do it) ... then you go back to factory default ... once again ... and you find out that it operates okay ... then and only then ... would I try to manually restore everything one by one. Your pictures are probably already backed up (maybe on your computer, maybe only on iCloud ... I don't know). You don't have to worry about your e-mail, as that is all on someone else's server. Any "notes" ... you can reconstruct yourself, if you need to (you can actually e-mail out all your notes, if you want to). Any individual apps that have personal data ... well, maybe you might have to e-mail that info off to your e-mail account, upload it to Dropbox, or manually write it down.
But, I really don't think you'll need to resort to all this "manual restoration" ... it's just your "last stop" if nothing else works. And ... yes, all your apps (that actual apps themselves and not your personal data on the apps) will all be available to manually download them once again.
Hopefully you'll simply find that you can restore your personal data from your backup and everything is okay. That's the preferred outcome ... :-) ...