If you operate online with customers around the country, you may be deemed to operate in every jurisdiction in which you have customers. That could get expensive if you need to qualify in each jurisdiction. I would definitely consult an attorney on this issue.
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You do not have to register in the state where you live. However, you will need to file for authority to do business in the state where you will be located. For example, you can file a Delaware corporation and file for authority to do business in NY (or the state where your company will be "located"). Even though you may not have a "location" per se in NY, I will assume that any employees that you have and any calls the entity would receive would wind up in NY so you are "doing business" in NY. You will also want to make sure your terms and conditions are properly drafted to include the right venue in the event of a dispute.
You are free to domicile the entity in any jurisdiction you like, but if you are performing business activity in NY, for example, then NY requires that you qualify the entity there as well. So, there may not be any meaningful benefit to registering the entity outside of NY if everything is taking place in NY to begin with.
The DBA filing in NY is at the state level. In most states it is done at the county level.
I will link you to a brief overview on the topic of foreign entity qualification and suggest that you reach out to a lawyer (or several) for a free phone consultation so you can get some specific insights on your particular situation. Getting the proper guidance upfront can go a long way to avoid painful issues later.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed with the law firm of Natoli-Lapin, LLC on the basis of this posting.