In general, what types of Cassegrain telescopes are there, and what advantages do they have over other types of telescopes?
The main reason is that Cassegrain telescopes are much shorter than Newtonians for the same focal length. This means that the overall cost of the observatory, which includes a mount, dome, and building, will be much lower.
A secondary reason is that professional telescopes use heavy detectors at the focus of the telescope. Newtonian telescopes put the focus on the side of the telescope, so heavy instruments and detectors would need to be placed on the side of the telescope; in general, this can make things difficult, because the weight on the focus exerts a variable torque on the telescope's tube.
It's much easier to support these heavy instruments if they are along the optical axis rather than on the side of the tube.
Extremely heavy instruments like very large spectrographs sometimes require different, more specialized telescope designs like the Nasmyth focus.
As for your question on different types of Cassegrain - this is a nomenclatural issue and I'm sure there are many answers, but some I can think of: Nasmyth telescopes are sometimes called Nasmyth-Cassegrains. Occasionally, people lump Gregorians in with Cassegrains. Schmidt Telescopes use a spherical rather than parabolic mirror, which takes some corrective optics but provides a much larger field of view - as much as six to eight degrees, so they are used for surveys frequently. Like I said - I'm sure there are tons more.
The Wikipedia page is very informative:
There are two major categories of Cassegrain designs: pure catoptric (mirrors-only) and catadioptric (mirrors and lenses).
Parabolic primary mirror, hyperbolic secondary. Main advantage: it's very compact and short for its focal length. Used a lot in astrophoto.
Both primary and seconday are hyperbolic. Primarily used as an astrograph, because the star images are round, making it easier to determine the distance between two star centers. Difficult to make. Almost all of the huge professional telescopes these days are R-C.
Elliptic primary, convex spherical secondary. Easier to make than R-C. Star images are not round.
There's a glass corrector plate in front of the mirror, with a fancy shape, that corrects the aberrations of the mirror. The mirror is spherical. Used for astrophoto.
Spherical glass corrector meniscus. Spherical primary mirror. Spherical seconday, which is usually a silvered portion of the meniscus. Used for astrophoto.
Various other catadioptric systems using sub-aperture corrector lenses.
Cassegrain telescopes are classified by the type of corrector they use. Classical Cassegrains have no corrector, and tend to have long focal ratios, f/15 to f/25. Maksutov-Cassegrains have a thick, deeply curved spherical meniscus corrector. They are simple to make because all surfaces are spherical, but use up a lot of glass because of their deep curvature. Schmidt-Cassegrains have a thin aspheric corrector plate. Schmidt corrector plates are mass produced so that they are inexpensive, but are gennerally not of as good optical quality as Maksutov correctors. The advantage of all these types is that they provide compact, relatively well-corrected optical systems with long focal lengths. Recent modifications of the Schmidt-Cassegrain design have added correctors close to the focal point to improve sharpness over a wider field of view.
Most modern professional telescopes are not Cassegrains at all, but use the Ritchey-Chretien design. This looks like a Cassegrain, but the mirrors are figured differently: both primary and secondary are hyperbolic rather than spherical. The Hubble Space Telescope is an example of a Ritchey-Chretien system. This provides better correction over a wider field of view, but the hyperbolic surfaces are hard to produce.
What types of cassegrain telescopes are there?
The Wikipedia article makes a decent resource useful to enable further research; the list, though I'm no authority on whether it is exhaustive or not (and I would think not), starts with the 'classic' cassegrain and then a few adaptations (Ritchey–Chrétien, Dall–Kirkham), to catadioptric assemblies (Schmidt, Maksutov.)
In the case of the non-catadioptrics types, each of these seem simply to tweak shape and / or tilt of mirrors, whereas the catadioptrics combine technologies - neither assuming something different, but each fit for purpose (though, for whatever flaws some might seem to correct in the original design, regression of sorts can become apparent in the features (as mentioned in the linked article regarding the 'off-axis configurations'.)
What type are most professional telescopes?
Even with just these few, we see utilisation of various types of telescope - the variety itself is the benefit of the game here, the availability of such allowing us to select based on the MO; for instance, if all telescopes were X-ray types then we'd be missing out, yet that's no reason not to have one or some of that type.
Surely cost is going to differ from one to the other, but in building to such scales and purposes I'm not so sure this would be a deciding factor, and so don't think we can huge bring economical benefits into it. Lastly, harking back to manufacture: I do believe that one benefit of the cassegrain type telescope over that of the Newtonian and refractor is that the cassegrain can be constructed more compactly while retaining aperture. In turn, we can infer that this ought to make them cheaper.