If you don't control any permanents of one of the types, you'll still choose the permanents of the types you do control.

Cataclysmic Gearhulk will be sacrificed to its own ability if you don't choose it as your artifact or your creature.

A permanent with more than one type may be chosen as either of the types or both. For example, an artifact creature may be chosen as the artifact, the creature, or both. Choosing the same permanent twice this way has the same result as choosing it once.

Lands can't be chosen and won't be sacrificed, even if they have other types (such as an artifact land).

First the player whose turn it is chooses permanents they control, then each other player in turn order does the same. Each player will know the choices made by players who chose before them. All of the unchosen nonland permanents are then sacrificed simultaneously.