1. Is science a branch of theology, or is theology a branch of science?
The question implies that they cannot be mutually exclusive, and indeed they cannot, because if properly practiced, they both encompass the subject of the supernatural, and they both encompass natural phenomena.
The answer to the question is that the one which came first is the trunk, and the one which came second is the branch. Theology came first, and science afterward.
2. Now to answer an implied question: how can they be different disciplines, if they both encompass the same set of subjects?
The answer is that they have different foci. Theology seeks foremost the knowledge of God and His supernatural Kingdom, and it seeks knowledge of the natural as a means of better understanding that which is supernatural.
Science, properly practiced, does not ignore the supernatural, because, while its focus is on the natural, it takes note of the fact that the supernatural affects the natural. Therefore, an honest scientist will of necessity (and hopefully also of his own desire) not only acknowledge the existence of the supernatural, but also will attempt to understand, study, and draw conclusions about it.
Any “scientist” who does not do both of these things is a fraud.
There are a lot of “scientific” frauds out there today.
3. To which discipline do we defer on a question, when the two are in conflict?
My answer may surprise you. My answer is that we have to look to the Word for guidance. If neither discipline’s answer to the question is biblically sound, then we defer to neither. And if one is biblically sound and the other is not, we defer to the first — even if that one happens to be science.
Science and theology both seek (and properly so!) answers to the same set of questions, which is the set of all questions. And, if properly practiced, both use exactly the same method to seek such answers. The difference between them is quite simply their area of focus. Each can enter the other’s area of focus whenever necessary, even correcting the other when necessary, but they remain primarily focused on their own area. In this way, they can work together for the pursuit of the same ultimate set of answers — i.e., the set of all correct answers.
4. One final question for you. What foundation do each of these two disciplines have in their search for truth on the same set of questions?
It is the same for both: Jesus Christ. I Corinthians 3:11 .
 BUT note well that one cannot properly “correct” the other if the first is biblically unsound in its position, and the second is biblically sound.