If I edit out one of the obligatory fields (eg. [{ $oxcmp_user->oxuser__oxzip->value }] ) in the order.tpl what else would this affect?
The registration process would stop. First, ditch this field (oxaddress__oxzip) in admin -> core settings -> settings -> other settings -> Mandatory fields in User Registration Form.
That worked. However, editing out the field did not remove if from the form. What file do I need to change to alter the layout of the Billing Address form?
which place exactly do you mean? Maybe there are some places “affected”. I just tried it with register.tpl and it worked perfectly.
Field titles and user input order: In lang.php.