The Byzantine Catholic Church has all of the below except for the beard as well.
The bread issue—leaven vs unleaven was one of the bones of contention contributing to the schism between the Eastern(Orthodox) and Western Church in 1054.
The whole issue of priests or monks in the West having beards is rather hilarious. It was prohibited for most priests as a sign of vanity. However, some religious orders required/expected their members to wear beards (cf Padre Pio).