If Earth is fixed, does the universe rotate daily, and what about faster-than-light speeds or frame-dragging?

Yes, the whole universe rotates around a fixed Earth once a day. Seen from Earth, the galaxies must exceed the speed of light to do it, which general relativity allows, and the rotating universe itself produces the frame-dragging. The universe turns around a stationary Earth every 24 hours. Scripture teaches it, and Michelson-Morley and Michelson-Gale confirm the Earth is not moving. On the speed of light the answer is yes and no, depending on the vantage point. From the Earth's perspective, the galaxies must travel far faster than 186,000 miles a second to go around once a day. But in general relativity that is no problem. The speed limit belongs to special relativity. General relativity lets light and material objects exceed it, and modern cosmology relies on the same allowance when it teaches that the universe inflated faster than light. Frame-dragging fits too. Hans Thirring and Josef Lense showed that a rotating universe creates inertial forces, including centrifugal, Coriolis, and an axial force Newton never had. Those forces produce the frame-dragging. Hermann Bondi, Julian Barbour, Max Born, and Christian Moller all recognized that if everything is relative, general relativity must allow a rotating universe around a fixed Earth. One caution. In general relativity, space-time is a mathematical construct, not a substance. Quantum mechanics treats space as real, made of Planck-scale particles. That real medium is what rotates. Thus it allows Special Relativity to make space and time flexible (e.g., anything it wants). This is why they say that it is space that is "expanding" not material objects when they claim that the universe is expanding. In our case, the space and time of General Relativity are made small so that light can still travel at c. The reality is, of course, that the universe is bigger at its rim than at its core. So, the whole thing depends on how we measure. If light at c is the only absolute, then it has to serve as the only measuring device, and thus everything else has to adjust. What adjusts is space and time (to anything the Relativist needs to make his theory consistent).