Q: complete phone backup before repair

If you wish to backup all of your data, including your Health Data, you need to have an encrypted backup. iCloud will provide encrypted backup, but you need to understand everything that iCloud backs up versus everything that iTunes backs up. You may want to take a look a this support document about backups to understand all of that.  About backups for iOS devices - Apple Support

If you choose iTunes, then also take a look at this support document about encrypting an iTunes backup. Read the entire thing, because if you forget your password for the encrypted backup, you will have to delete it and restore the device to factory.  About encrypted backups in iTunes - Apple Support

The iTunes backup will backup your photos, but it is best to import them to the computer for safer keeping. As far as mail is concerned, depending on your account, it is saved on your server. Mail should be saved. If you have an IMAP or Exchange account, the mail is on the server. POP mail is the only mail that is downloaded to the device itself.

The both will backup contacts, messages and call logs, unless you are syncing your contacts via iCloud. If that is the case the iCloud backup will not backup the contacts, since they currently exist in the cloud, and would be available there to sync again with the new device. You would just log into the same iCloud account on the new device and click on contacts, and they would be synced to the new device.

The only thing I can say about contacts, is if the phone is the only place you have them stored, and are depending on the backup to make sure you still have them, you are asking for trouble. If something was to corrupt your backup, and then something goes wrong with the phone, the contacts are gone. They should be synced to either iCloud, another cloud service, an email account that provides that feature, or synced to a supported application on the computer, such as Address Book, or Outlook.