With the concept of a Shopify Development Store, they are meant to be completely free of charges (unless you go onto the paid plan). This means you cannot install any paid apps on your store, and you also wouldn't be charged for any payments/charges you make with your own app because they will be considered test charges.
In terms of how to test with the Billing API, as long as you can create the charge (e.g. ApplicationCharge, or RecurringApplicationCharge, or UsageCharge, etc.) successfully using the API for your development store (i.e. you get a 200 response from the API) and you can receive the e-mail confirming the charges, that should be enough. If you can also see your charge when you perform a GET request, even better. When you publish your app and start charging real merchants, Shopify will ensure your successfully created charges with the Billing API will appear on the merchant's billing invoice at the end of every billing cycle.