Who throws the dice for you?

[Angrboda]

1. If God exists, there will be unexplained behavior.
2. If God does not exist, there will not be unexplained behavior.
3. There is unexplained behavior.
C. Therefore God exists.

This seems an equivalent formulation with no question begging. Am I missing something?

You have to assume 1 and 2 in order for C to be true. There's no proof stating that "God" is the only explanation for unexplained behavior, or that unexplained behavior can't happen without God.

Picture it this way: you live in an ancient culture that doesn't know that the earth spins, which causes the sun to "rise" and "set", making this behavior unexplained:
1. If Apollo exists, he will ferry the sun across the sky in his chariot.
2. If Apollo does not exist, the sun will not move across the sky.
3. The sun moves across the sky.
C. Therefore, Apollo exists.

Do you see the problem there?

I see you have a problem with the soundness of certain premises, however begging the question relates to the validity of the syllogism, not its soundness. In premises 1 and 2, the existence of Apollo is contingent, not assumed, so I don't see how either premise is an example of begging the question.