It doesn’t. It reads only a tiny portion of the disk at any given moment. The disk spins and the reading system slowly works its way from the center of the disk towards its edge, following a spiral path around the disk.
Optically, they are identical. However, the laser disc player uses an analog recording technique (non-digital) to recreate the video signal. I think that the lengths of ridges (or perhaps pits) in the aluminum surface are used to control the analog signal strength.
The optical system of the CD player must be very clean. If the final lens has dust on it, the photodiode will not see the full range of light and dark patches that it expects. A cleaning CD presumably cleans this final lens, although I’m not sure how. In principle, the whole CD player should be pretty resistant to dust problems because the laser beams are large except when they focus on the CD itself.
I only know how the prerecorded mini disc players work: they work a lot like CD players. However, they use a much smaller disk, made possible by intelligent data reduction. Instead of using 16 bits to represent each current measurement, the mini disk uses a variable number of bits. The recording equipment determines how many bits are needed to represent the sound accurately and eliminates unnecessary (or inaudible) details in the current measurements. The optical systems in mini disc players are the same as CD players.