Read Apple's Terms and Conditions
If there is nothing specifically in the terms and conditions that prevent someone your age from signing up as an Apple Developer, then you shouldn't be breaking any rules or laws. You may need help paying the fee; however, since you can't legally apply for the credit card you would need to pay Apple's developer fees. Since you've already done this, it sounds like that's not a problem.
In regards to Terms and Conditions, i'll use email as an example. It would violate Google's terms and conditions for anyone under the age of 13 to create an email address. Therefore, even a legal adult cannot legally create a Google account for children under age 13. If Apple's Terms and Conditions have any age restrictions in them, then you could have problems later on when it comes time to cash out your earnings.
As long as there is nothing explicit in Apple's terms and conditions restricting someone your age, you're probably okay. Probably...
Finally, it's worth noting that this is not a site for legal advice. I'm not really qualified to give legal advice. Nothing you read here will hold up in court. The best thing you can do is talk to an attorney.
Starting a Business - Contract Law
In regards to starting a business, that is a completely different matter. You'll need to check with the laws of your state or local government. Most likely, you'll need a parent or guardian just because there are many contractual obligations that you will not be able to enter into because you're under 18.
UPDATE: You can enter into a contract. However, many entities may be hesitant or unwilling to enter into a contract with you since, by law, you as a minor can merely disaffirm it.
Do you really need a business name?
Starting a business can be a very costly and time consuming process. So if you don't have any employees, then perhaps you should ask yourself if this is really necessary. If this is your first app or you haven't made any real money yet from your work, then I would suggest you focus more on that aspect of your business. In other words, make sure you're actually going to need to go through the process of starting a business.
Good luck on your iPhone App!
I'm not an attorney. You shouldn't rely on my advice in this matter and should talk to an attorney who specializes in matters such as this.