Can I do a factory reset without losing my pictures and apps? Google Play won't let me download anything, search anything, or log in. It keeps giving me an error saying I have to log in. Everything but Google Play is working just fine. I want to reset my phone but most people are saying I should download an app to keep all my info save so I don't lose anything but that's the issue. I can't download anything. So it's kind of just a circle of hell and I don't know what to do. Should I just do a factory reset and hope for the best? I've gone into my settings and checked so that everything will supposedly be backed up but people are saying it's untrustworthy and when I go to do a factory reset it says it will delete everything including the data I've chosen to backup. Should I trust it?
A factory-reset has exactly that purpose: to "undo" all changes the user did to the device, and bring it back to a state as if it just left the factory. Which means, all apps you've installed and all your apps' data will be removed. What might remain is what's stored on SD card (unless you're using adoptable-storage or your device's manufacturer decided to include it with the purge anyway.

As you're already having issues logging in to Google services, I'd not trust the "Google Cloud Backup" being anything like complete. And even if it were, you never can be sure it will restore, or restore completely: the user has no way to control that. So no way to trust this either.

You could do a backup using ADB (see e.g. Full Backup of non-rooted devices). But as the Playstore app is the only one having trouble (does Gmail work?), I'd rather focus on that. Check our google-play-store tag-wiki, where we've linked a bunch of troubleshooting questions – Getting “Authentication is required” when trying to access the Play Store seems to match your issue. Also note that sometimes this is just a temporary "hick-up" which solves itself within a few ours (or sometimes days).