Most states do DBA registrations (also called "trade name" registrations) at the state level with the secretary of state of that state. California is among them. (Colorado transferred DBA registration responsibility from its Department of Revenue to the Secretary of State a few years ago.)
Generally you would have to do a trade name registration in every state or country in which the company does business under that trade name.
You would also need to register as a corporation authorized to do business in every state (other than the state of incorporation) where it does business. This simple registration document lists the true state of incorporation and designates a registered agent within the state where it does business, and would list of the principal office of the corporation worldwide and also within the state of registration. Sometimes these forms also allow the foreign corporation to list a DBA used in the state where it does business.
For example, Kroger, which is based in Ohio, operates under different trade names in different states: it is called "King Soopers" in Eastern Colorado and "City Market" in Western Colorado. It would have to file trade name registrations for King Soopers and for City Market, respectively, in Colorado, in addition to a registration of a foreign corporation in Colorado. There would be no harm in filing a trade name registration for these names in Ohio as well, as that would protect its right to rebrand using its Colorado names in Ohio at some point in the future and the cost of the filing is negligible for any decent sized business.
It would be unusual for states to handle trade name registration at the county level, but I wouldn't be surprised if there are a few that still do so.
If you sell retail goods or some other kind of goods subject to an excise tax, you will probably also have to get a state and/or local license related to the collection of those taxes.