If the FREE SHIPPING module is active and shown to the customer during checkout, all other modules automatically disappear. That is standard ZC logic. You either offer Free Shipping or you don't (based on certain pre-determined criteria).

To enable free shipping for customers and at the same time offer other shipping options to customers, you need to use the Free Shipping Options module instead.

The Free Shipping Options module allows you to specify different conditions for a customer to qualify for free shipping. For example, when the total order exceeds a certain amount or weight, or when the total number of items bought is above a certain threshold value.

This 'Free Shipping Qualifier SideBox and Shopping Cart Module' has not been tested to work with the built-in Free Shipping Options module.

You may want to do some testing yourself and post your findings here.