An LLC registered in California will be required to have a registered agent. You can begin the legal process by serving a demand letter or a summons and complaint on the registered agent.
I would consult with a business litigator in your area to get a better understanding as to how this process will play out.
If you are serious about this business contact a good incorporation attorney in NYC. They will be able to walk you through the entity options as well as discuss the license requirements and other issues (such as insurance/licenses/etc.)
Forming a company is relatively simple and affordable, but it is good to get counsel to help make sure you have addressed all the issues.
If you registered your LLC you should have the original incorporation documents, records of having paid the incorporation fee, and be on file with the registered agent. Should be pretty simple to prove you were the founder if anyone ever tried to claim ownership of your entity.
I would contact a business attorney who handles trademark matters and make sure that you fully understand how the process is playing out. Section type 1(a) is an application based on actual use and 1(b) is a filing for intent to use. That could be the issue but without more information it's impossible to tell.
Your question blends incorporation and insurance issues. For a professional practice both are important. I would find a good business attorney who can sit down with you and review which combination of entity structure and malpractice coverage will go the farthest in protecting you and your assets.
Just based on the complexity of your inquiry I'd recommend that you contact an attorney that deals with nonprofits regularly. Given the regulations that surround how nonprofits are governed you will want to make certain that your response to this issue is measured and based on informed legal opinion. Someone will have to look at the documentation (agreements/emails/etc) between your organization and the other nonprofit and figure out what representations were made, etc.
If the company will just be you then you can get started with a very simple LLC. If you have grand plans of growth and bringing on investors, etc. - then a corporation may be the route. Both offer protection from personal liability but there are slight tax differences that should be discussed. I would contact a lawyer who specializes in formation and incorporation work and sit down with them for a consultation.
By establishing a NY address you will likely have to register your company as doing business in the state of New York. This would have initial costs as well as tax consequences for future revenue and could subject you to jurisdiction in NY courts. I would contact your accountant and a good business attorney to fully discuss the issues associated with establishing your business in another state.