Nationwide sales rep and LLC
1. Should I form a LLC to enter into contract with this agency?
2. Where should I file the LLC? If so, CA, WY or Delaware?
3. Is this considered as 'Doing business in California' when my client (who pays me) is in New York?
I would generally advise you to form an LLC in order to separate the business liabilities from your personal assets. In your case the risk might not be very high, but its still better to separate whatever risk there is from yourself.
With that being said the question is where to form the LLC - in California or elsewhere. California rules are pretty aggressive in the definition of "nexus", meaning most business activities would create a nexus in that state, which technically requires you to register for tax. So if you choose to form an LLC and most of your activity will be in California you should choose California as your state of formation.
Of course, maintaining an LLC in California is costly, as the minimum LLC franchise tax is $800 a year, so if you think that it's too much you could start as a sole proprietor with or without a DBA and then consider forming an LLC as the business progresses. Keep in mind, if you register a DBA as a sole proprietor in California there is a DBA publication requirement in the county of registration.
Finally, your question about "Doing Business As" in California is actually a question of whether or not you create a nexus in California while being paid by a client from New York. The answer has to do with the type of activity in California rather than who pays you. For example if you solicit businesses in California to purchase a product you create a nexus. If you maintain an office in California you create a nexus. And the list goes on.