I see following notice on almost every checkout page (e.g. billing step):
Notice (#1): Undefined index: cust_shipping_addr_block in ...\core\kernel\db\dbitem.php on line 273
I'm sure, that there is shipping_addr_block custom field defined in user configuration. In fact you can even see and change it in Admin Console.
It seems, that code, that should be creating custom fields on Front-End is blocked by "recursion protection code". This might as well prevent any type of custom fields from working normally.