Our Juniper hedge is 12' tall and 6' wide. Can it be cut down to 5' tall and still survive?

That is a severe reduction and we would probably not recommend it without first seeing the hedge. Depending on the species of Juniper, an acceptable reduction is usually around 20%, which would mean approximately 2.5 feet, although again depending on the species this amount could be increased. You should make an appointment for us to see the hedge and assess the possibilities of reduction.