Ability for a customer to enter the company they represent in the user profile, which would also appear on invoices.

I just found that you can enable a Company field on the payment page, so I assume it will show up on the invoice, but I apparently it doesn't save that information in the customer profile…

Hi,
 
The fields for a user profile can be edited at 'Administration > Customers > Profile Settings / Types > Profile Settings'. There you can check/uncheck checkboxes for each field available on a user profile page.
 
Regards,
ViArt Support Team