I am helping a new faith-based non-profit organization get started, and I'm dealing with their incorporation issues.

Are you an attorney or tax professional? I would be careful regarding giving legal or tax advice to an entity if you are not qualified to do so. I do not know the answer to yoru first question as I have never researched it, but at the very least you'll need a registered US office to incorporate. Most non-profits, depending on state law, register as non-profit corporations. Obtaining that is simple but is just the beginning. Again, depending on what state you choose to form your entity in you will need to register with the state charitable organizations office and if you would like donors to be able to deduct donations make to your organization you will have to file as a federal 501c organization using IRS Form 1023. Starting a non-profit is much like starting any business - there are legal requirements and it costs money and takes time therefore experienced advice is an absolute requirement. In addition running an non-profit properly is actually oftentimes more difficult than other for-profit businesses because you are held to a high standard thus having proper board governance procedures and things like conflict of interest and non-discrimination policies is vital. I am not particularly a non-profit attorney but I have worked on boards and many an attorney has told us - if you do not have the money and time to form properly you should not begin work as a non-profit.