You can link profiles created through regular CIM calls, which means you can initialize some of the fields - all if you include credit card data. But you can't pass data directly to hosted CIM forms. As for choosing fields, you can do this globally through your account control panel, just not for individual forms. I've copied and pasted from my control panel, these have checkboxes next to them for View, Edit, Required.
Payment Information
Recurring Billing Transaction
Card Code
Order Information
Invoice No
Description
Customer Billing Information
First Name
Last Name
Company
Address
City
State
Zip Code
Country
Phone
Fax
Email
Customer ID
Customer Type
Customer TaxID/SSN
Drivers License Number
Drivers License State
Drivers License Date of Birth
Shipping Information
First Name
Last Name
Company
Address
City
State
Zip Code
Country
Additional Information
Tax
Freight
Duty
Tax Exempt
PO Number
Security Code
This security feature helps to prevent unauthorized access to or abuse of your Payment Form by automated Internet Web programs. When this feature is enabled, the customer is required to enter a security code from a randomly generated image embedded in the Payment Form to continue with their order.
Require the Security Code feature on the Payment Form.
Shipping and Return Policies
Include a link to your shipping and return policies page on your Web site on the Payment Form.
Shipping and Return Policy URL: