Do I have control over which fields on the billing page are mandatory and which fields are optional?

Yes, you have complete control over all fields in the billing and shipping addresses. On the Commerce Setup > Order System > Shipping Screen, you can specify whether each field is displayed, whether they are required, and how many characters each field will accept. On the Store Preferences > Store Text > Checkout screen you can set the text use for the field labels, such as "Name" and "Address 1."

If you have ShopSite Pro or ShopSite Manager and you have enabled UPS Address Validation, customers must enter a valid city, state, and zip code combination or the order will not be accepted.