“Thy” and “thine” are the equivalents of “your” and “yours”, respectively.

That’s partially correct. The second part, that thineis equivalent to yours, is correct. But the equivalent to your is either thy or thine, depending on whether it’s followed by a consonant or vowel sound, respectively.

BTW, my and mine used to work exactly the same way. For example: “Mine eyes have seen the glory of the coming of the Lord.”

A Web search should turn up many sources for this information.

No, you’re thinking of thou (subjective case) and thee (objective case).

Thy, not thine.

Same rule as a/an: the first one is used before a consonant sound, and the other one before a vowel sound.

I’ve always associated it with the classic “riding into the sunset” storytelling trope, which is often used to symbolize the end of a journey or of a chapter of the rider’s life.