I think you’re confusing a payment gateway with a merchant account. A merchant account is a back-end account that facilitates transferring from a credit card to your bank account. It has no interaction with a customer. A payment gateway is what would integrate with your website or what would allow your customer to enter their information. Most do not store credit card numbers at all but some offer secure storage vaults which allow for future or recurring billing.
As far as shipping only to a billing address, this is something that you would typically do in your shopping cart itself. You would configure your checkout system to allow a shopper to only use 1 address for shipping and billing. This is a decent way to combat fraud assuming that AVS and CVV are both available. However, you can expect a significant reduction in customer conversions. Requiring a single address will prevent anyone with a gift card, or non AVS supporting card (basically anyone not in the US) from making a purchase. It’s also common for customers to ship to their business and bill to their home and vice-versa.
The payment gateway and merchant combo that has worked well for our stores is Authorize.net and Gravity Payments. Authorize.net is compatible with just about every cart out there, and Gravity Payments has been a fantastic merchant provider. Their rates are pretty low (around 2.0% with a volume discount), and their phone service is impressive. Highly recommended.
As was previously mentioned, the AVS settings can be set in the cart control panel.