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Yes, you need to register your business with your state -- the process depends on the business structure you select and which state you're in. (Actually, I'm not sure how you were able to set up a business bank account without having done that already.) If you can provide some more information about your business, your location, and the steps you've taken so far we can probably offer more specific guidance.

No, you do not copyright a business name. Registration provides/ensures uniqueness in that jurisdiction to some degree (and you obviously can't give your business the same name as another business in the same industry/market), but issues beyond that such as trademarks are a few steps ahead of where you probably are at this point. Hope this helps, and that you can provide some additional info. Best wishes.