If a permanent is sacrificed to pay a cost of a spell or ability, Mayhem Devil's ability will resolve before that spell or ability. Conversely, if a permanent is sacrificed during the resolution of a spell or ability, that spell or ability will finish resolving before Mayhem Devil's ability is put onto the stack.

A legendary permanent that is put into a graveyard because of the "legend rule" isn't sacrificed.

If you sacrifice Mayhem Devil, its ability triggers.

You control Mayhem Devil's triggered ability and choose the target, no matter who sacrificed the permanent.

Mayhem Devil itself doesn't allow any player to sacrifice any permanents. Its ability triggers whenever a player sacrifices a permanent because some other spell, ability, or cost instructed the player to do so.