Does Apple send an alert prior to the 3 month trial ending for Apple Music?

No, but you can set one up yourself using Reminders (or Calendar, if you prefer).

I get email notifications when subscriptions are about to renew or expire. I don't recall if that applies to the trial periods of subscriptions, though.

No. It will auto-renew without notifying you about it.

It's up to you to cancel it before it renews and starts charging you.

Manage your Apple Music membership - Apple Support - https://support.apple.com/HT204939 - "Learn how to change from an Individual membership to a Family membership, turn off automatic renewal, or join Apple Music later." Turn off autorenewal and your membership will finish at the end of the current subscription period.