My customer export does not include addresses or customer groups. Is something wrong?

No, that is what the export contains. Export CSV on Customers → Manage Customers downloads every customer (deleted ones excluded) with these columns: id, name, email, phone, country code, status and registration date. Passwords and other sensitive data are never exported.

Addresses and groups are not in the file. To see a customer's saved addresses, open them and go to the Addresses tab. For buying history, use the Orders tab, or Analytics → Reports for customer reports such as Customers by location.