First and foremost: provided that you trust 3rd-party APK provider sites (APKPure, APKMirror, etc.), no Play Store access is not a problem - you will only be restricted from paid apps. For trust issues read this question (and others related) on the site.
Contacts:
- In the Contacts app, top-right menu -
import/export - export to .vcf file.
- Select items you want exported.
- A .vcf file will be saved by default to the root of the internal storage - back it up somewhere.
- Import later with the
import from .vcf file option.
SMS: use the app SMS Backup and Restore. Been using the Pro version of the app since Android 4.x and it always worked smoothly. Backed-up content will be stored in the SMSBackupRestore folder under internal storage. APKMirror
Mail: your mail is supposed to be all up in the clouds, i.e. synched with your mail server, so there should be no need of backing them up in the first place.
Media files (picture, music, etc.): provided that you know the location, just copy them off the device. Use the details option in the Gallery app if you need help locating a specific picture.
App data: the free version of Titanium Backup is enough for backing up individual apps' data. Backed-up content will be stored in the TitaniumBackup folder under internal storage. Note that, before restoring app data, you might have to start them once. APKMirror
Copy backed-up contents off the device (keep folder structures intact), then after flashing LOS, copy them back in place, and use respective restore options.