Do the these missions have nicknames?

Santa Cruz does not have a generally recognized nickname. It has been referred to as the "hard luck" mission because the original mission was flooded and had to be relocated and the new mission was sacked by the occupants of the nearby town during the Pirate Raid on California.

San Antonio de Paduais referred to as Mission San Antonio.

San Buenaventura is called the Mission by the Sea.

San Carlos Borromeo de Carmelo is known as Mission Carmel.

Santa Clara de Asis is often abbreviated to Mission Santa Clara.