Can I do a factory reset without losing my pictures and apps?
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).