With most rolling shutter cameras, at least those intended for film and TV work as opposed to webcams and cellphone cameras, there is a period of time during which all of the photosites on the sensor are sensitive to light which will become part of the final exposure. There is then a generally much shorter period in which the rows of photosites are read out sequentially.
Yes, it's an equivalence to how a 35mm film camera would work. There is no angled disc involved, but come cameras refer to shutter angle because it's a familiar terminology.180 degrees implies a 50% duty cycle, that is, 1/48s exposure at 24fps, or 90 degrees implies a 25% duty cycle, or 1/96s at 24fps. The digital camera is using sensor timing to create those exposures rather than mechanical parts (except in a small number of cases), but the result is similar or identical.
This simple equivalence is (albeit very slightly) inaccurate in the case of rolling shutter cameras where some pixels will be active for longer, or during a different period, to others. Other than in very specific circumstances this will not make any practical difference. I'm not sure if there's a convention as to whether the per-frame exposure time of a digital cinematography camera includes the readout period, or perhaps some part of it, or none of it at all. Either way, in many cases the readout period will be thousandths of a second whereas the entire frame's exposure will be many tenths, so the difference it makes is tiny.
The use of shutter angle terminology does not relate in any way to the potential for apparently skewed objects as viewed through a rolling shutter camera during horizontal motion. That's an entirely unconnected phenomenon.
CMOS sensors are used because they make it easier to build electronics other than just the photosite array onto the sensor itself. Modern sensors require multiple "taps," that is, data pipelines to get the picture information off the sensor quickly enough to work properly. This means that various areas of the sensor will be individually served by output amplifiers, digital to analogue conversion and other electronics. It is not possible (or at least not effective) to try to build this with the sort of manufacturing processes used for CCD sensors.
CMOS sensors can have rolling or global shutters. All things being equal, the global shutter is preferable because it creates fewer problems (flash banding, skew, and so on). The Blackmagic 4K cameras have a globally-shuttered 4K sensor. The problem is that to create a globally-shuttered CMOS sensor, each photosite on the device must have (at least) one extra transistor built into its design to control whether or not it is sensitive to light. This is a disadvantage because we want the sensitive face of the device to be filled as much as possible with light-sensitive area, not electronics, to maximise sensitivity and dynamic range, and minimise noise. It also makes the sensor harder to manufacture. Future technologies based on layering electronics and light-sensitive components together may make it possible to produce globally-shuttered sensors without these compromises, and we're all looking forward to that.
The simple solution is to use a mechanical shutter just like a conventional film camera, which is what the F65 and certain Alexas do, but of course the mechanical engineering is expensive.