Hi when you factory reset, it will wipe all settings to default as if you bought it brand new from the store (this is the intended result anyways). So it's likely to reset what was causing the black and white issue so long as it's software related. All of your apps will be uninstalled also. If it's a hardware failure this may require a repair shop.
As for being logged in, it allows you to sync your google data and back it up into "the cloud". For contacts, gmail, google photos etc. I suggested logging in before hand so that it could do that.
All you would have to do is log back in after a factory reset to access the same stuff as if it were never gone.