Does my Apple music subscription allow me to purchase songs at no charge

No, but it does allow you to listen to anything in Apple Music while you have an active subscription. See: https://www.apple.com/apple-music/

Whether or not you are subscribed to Apple Music you can still buy music from the iTunes Store. See: Buy music from the iTunes Store - Apple Support

No.

Continuing the subscription after the trial will require payment; if you choose not to continue, you won't be able to play anything you've downloaded from it.

If songs or albums have individual prices listed, they're not part of the subscription and will need to be paid for separately. They're one-time purchases.

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