There's a lot of loosely connected questions here, so let's go through them one by one.
Which rotational frame does the universe prefer?
The universe doesn't prefer any frame. There is no "universal" reference frame. This is one of the principle axioms of Special Relativity. You cannot pick a frame and say this is the "correct" frame. The reason being that you have no means of establishing what "correct" even means. One person may claim their chosen frame is correct, but another person might choose something different and claim theirs is correct. Neither person would be able to show their frame is better or more "correct" than the other's.
That being said, I think its important to point out that while no true universal reference frame exists, scientists sometimes use a de facto "psuedo-universal" reference frame. That frame is the one which is at rest with respect to the Cosmic Microwave Background (CMB) radiation. It is not a true universal reference frame, but because of the importance of the CMB, the frame at rest with respect to the CMB has been given more (arbitrary) importance by scientists, hence the term psuedo-universal. You can read more about this at this great physics Stack Exchange question.
We say that the earth rotates once per 24 hours. But is this the master frame?
This is a good question. Anytime someone mentions motion, you should always ask, motion with respect to what? If I'm inside a car, I would say the car isn't moving at all because my motion, with respect to the car is zero. A person standing on the side of the road would say the car is traveling at 25 mph. So you can see that defining motion requires defining a frame to measure that motion in.
When it comes to talking about the rotation of the Earth, there are actually a few different frames that scientists use. They're all very similar and produce similar answers (all about 24 hours). I'll restrict myself to talking about the frame which defines the Solar Day (another example would be the Sidereal Day).
The Solar Day is the time it takes the Sun to return to the same position in the sky as from the previous day. As you can hopefully see, this measure of a day is fixed to a frame with respect to the Sun. It would be the same regardless of the motion of our solar system through the universe and thus wouldn't be considered a "master" frame (not that there is one).
What is the solution to Newtons Bucket Problem?
This, I would say, is really outside the realm of the Astronomy Stack Exchange and really should be its own question, preferably on the Physics Stack Exchange. For that reason, I won't answer this question here, but I will point you to this great source which talks about the Bucket Problem is detail.