I understand that you always want to use a business banking account for your LLC lest you lose legal protections by intermixing personal money with business money.
Does the same apply if I were to open up an entirely new and separate Bank of the Internet personal account and use that to receive direct deposit payments?
It's difficult to say, but just open a business account. You should get a tax ID number for your LLC and should use that tax ID number to open a separate account for the LLC. Why take the risk?
I established Donohoe Digital LLC in Colorado and have moved two different times, now living in Illinois. I do not pay taxes in Colorado simply because my LLC is registered there. However, the laws vary from state to state. In cases like this, it is best to consult a professional (fortunately my brother is a CPA specializing in state and local tax laws).
In any case, it is wise to have a separate account for your LLC to keep your business expenses separate from your personal. You should think about the expenses your business incurs - apple developer program, domain names, hardware costs, software purchases (tools, etc.), legal fees, etc. All these items are legitimate costs of running your business that offset your income (which ends up reducing your tax liability). It is much cleaner to receive income and pay business bills from your business bank account. I also have a business credit card for these reasons and keep track of all this using QuickBooks. In the end, run it as a real business and keep good records. You'll be happy you did.
User6492 is correct that any bank account created for your LLC should be opened using your federal tax id and not our SSN. It is easy to get one (http://www.irs.gov/businesses/small/article/0,,id=98350,00.html).