The answers to your questions vary depending on a few factors, such as if you are conducting your business remotely, or through a physical presence such as an office or conducting your business in person.
In short, if you are conducting your business only remotely, you could theoretically register in any state, such as Wyoming. Keep in mind, if you are using your home address as your business, and you planned on writing off any business expenses, you would need to be registered in your home state.
If you have a strong physical presence (nexus) in a state, then you need to be registered in that state specifically. I'd like to point out a past question answered by our tax expert which explains how nexus is created which I believe you will find helpful.