First, I'd suggest minimizing your coding responsibilities. A manager is not a programmer, and it's cruel to ask someone to do both jobs. It sounds like you really like to code, however, so there's no reason to get rid of it entirely. But choose one to be your focus, and it sounds like it's being a manager.
Second, set up a system where you set yourself to Do Not Disturb. Let the relevant people know. If you're being interrupted physically, set up an obvious physical marker (like a red sign or a flag). If it's online, shut off your messenger and email. If it's by phone, turn it off... you get the drift.
Third, suggest that they help each other. I've never liked bureaucracies; they shouldn't need to go through you to each other. This is basically an extension of the Solution #1 you proposed, so make them more independent and interdependent.
Fourth, and this is a bit of a risk, is to permit them to get through to you "in an emergency", which is basically, "No one knows; Google is useless; I have nothing to do for the next hour." When that happens, you talk to them, sort it out, and then make the rounds to make sure no one has something pending, then go back to your job. Make sure that they actually
have checked with everyone else and Google, and if that doesn't help, they should work on something else until you're free.
Fifth, make the rounds every now and then. If you deal with their questions on your schedule rather than theirs, it's easier on you.
If you're technology heavy, then you might consider setting up a simple question-queue system, though that's dangerous, since it's asking for trouble.
