What exactly does iTunes backup with a full backup?

The music on your phone is not part of the backup. You can re-download any purchased music from the iTunes Store or re-sync the music to your phone from iTunes. Backing it up would be redundant.

Apple are not part of the backup, but in app data is if the app developer coded for that.

Pointers to apps are saved and apps will be re-installed from the App Store as part of the restore process.

See this article: https://support.apple.com/en-us/HT204136#:~:text=An%20iTunes%20backup%20doesn%27t%20include%3A%201%20Content%20from,Apple%20Pay%20information%20and%20settings%20More%20items%20

Backups from your computer

A computer backup of your device, which is not the same as a sync, includes almost all of your device's data and settings. A backup from a computer doesn't include:

• Content from the iTunes and App Stores, or PDFs downloaded directly to Apple Books
• Content synced from Finder or iTunes, like imported MP3s or CDs, videos, books, and photos
• Data already stored in iCloud, like iCloud Photos, iMessages, and text (SMS) and multimedia (MMS) messages
• Face ID or Touch ID settings
• Apple Pay information and settings
• Apple Mail data
• Activity, Health, and Keychain data (to back up this content, you'll need to use Encrypted Backup in iTunes.)

This Apple link describes backups ➡️ About backups for iPhone, iPad, and iPod touch - Apple Support

To address your specific question, the backup does not include anything that you sync to iCloud or anything that can be redownloaded. Thus, it does not include things like apps (which can be redownloaded from the app store), music, which can be resynced from your computer or, for iTunes music that can be redownloaded from the iTunes store. If you have an Apple Music subscription or you subscribe to iTunes Match for music that content will be redownloaded.

However, the process is pretty much automatic; you don’t need to do these steps manually, if you follow these instructions everything will be restored to the new phone ➡️ Transfer data from your previous iOS device to your new iPhone, iPad, or iPod touch - Apple Support

If you buy a phone either from a carrier or registered with a carrier it will come with a SIM in it. When you turn the phone on it will transfer service to the new phone.

Because it transfers the phone number. If you don’t transfer the SIM you will have to get a new one from your carrier. If you are upgrading to a 12 series iPhone you should get a new SIM to support 5G.

No, your playlists are app data for the music app, so you won’t lose them.

The SIM card has no data on it at all.