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Hi there, my name is Michael and I'm a business law attorney with experience in all aspects of business law. I'll do my best to assist you.
There are no special requirements to register an adult website.
In the US you can use any standard domain and register it as you would with any other website or business.
You real concern should be your home state where your material will be distributed and hosted. Make sure you comply with any regulations regarding indecency and the like.
A great resource for you is found here: http://www.adultweblaw.com/
That also includes examples of any disclaimers and forms that are often included.
Depending on how much control you will have regarding the production, sharing and distribution of the material, you should take proper precautions, including review and approval of content, and strict adherence to prevent any material from being "obscene".
The fact is it is a very risky area of business, and you face potential charges from any prosecutor that might view your material as "obscene", and the legal definition behind that, so again, be careful.
I hope this answer has been helpful. I'd be happy to further clarify my response and follow up with any other questions. If you're satisfied with my answer, remember to hit the "accept" button. Bonuses and positive feedback are always appreciated! -Michael