If you gain control of a Loyal Cathar owned by another player and it dies, it will be put into to its owner's graveyard and then return to the battlefield under your control.

Another double-faced card that becomes a copy of Loyal Cathar and then dies will return to the battlefield with its back face up.

When Loyal Cathar dies, it will return to the battlefield with its back face (Unhallowed Cathar) up. It doesn't return to the battlefield and then transform.

If a card that isn't a double-faced card becomes a copy of Loyal Cathar (due to Cytoshape, for example) and then dies, that card won't return to the battlefield. It will remain in its owner's graveyard.