Tate's rigid-analytic geometry was the first theory of (global) nonarchimedean geometry to have been devised, and in some sense could be seen as a "proof of concept" that such a theory can exist, despite some general skepticism which was present at the time, including from Grothendieck. For a while this theory was the only one, and when alternatives were introduced, they didn't present enough benefits for people to switch over to Berkovich or adic settings. You can see this question of mine where this is discussed in the answers.
Now, rigid analytic geometry is somewhat limited - you should think of it as analogous to the classical theory of varieties over fields. We would like to have a theory which is more akin to that of schemes, which could admit a greater range of spaces one can study. The fact rigid spaces are also not "honest" topological spaces is also often undesireable. Berkovich was the first to get around this, and his theory has a few advantages: firstly, one has a real topological space, and even a rather "nice" one - almost of "Euclidean" type, which makes it amenable to the usual topological study. Secondly, it is not strictly tied to the nonarchimedean world - one can apply it to any Banach space, including archimedean ones, which leads among others to the theory of Berkovich spaces over $\mathbb Z$ as developed by Poineau. Berkovich's original motivation, though, was that the bigger set of points gives a susbtitute of generic points, making it useful in developing the theory of etale cohomology.
Adic spaces were also developed by Huber largely with refining the theory of etale cohomology in mind. They are more tied to nonarchimedean algebras, and for those they can be seen as a further refinement of Berkovich spaces. This class of spaces permits you to use a much bigger class of rings, most famously perfectoid algebras, as shown by Peter Scholze.
Now to try and answer your questions:
Every rigid space has both a Berkovich and an adic analytification, which are compatible in the obvious way with analytification of schemes of finite type over a nonarchimedean field. There isn't an analytification from Berkovich to adic spaces in general, but there should be one for spaces topologically of finite type over a field, again compatible in the obvious way.
The statements should be the same, but the notion of coherent sheaf itself is significantly more subtle, especially in the rigid setting. This is discussed in Bosch's "Lectures on Formal and Rigid Geometry".
As I mentioned, studying etale cohomology has driven Berkovich and Huber to describe their theories. I believe this cohomology makes sense in all settings, and would expect suitable comparison theorems, but I'm not certain on that. In a different direction, let me also mention rigid cohomology, which is a cohomology theory of varieties in characteristic $p$ which is defined in terms of rigid spaces. It can also be defined using the other types of spaces.
Let me refer you once again to my other question on the use of rigid spaces - my general impression is that it can be still useful conceptually and pedagogically, and has the biggest backlog of literature written in this language, but can overall be replaced by the other theories. As I mentioned Berkovich spaces are convenient if you want to mix in the archimedean setting, and they have also found uses in dynamics and tropical geometry (IIRC in some sense, Berkovich analytification is the inverse limit over all tropical models of a variety). Finally, adic spaces are probably the best fit for most applications dealing with "nonclassical" nonarchimedean spaces, especially non-noetherian ones like perfectoids.
By the way, on those and related topics, I highly recommend the set of notes by Brian Conrad on Several approaches to non-archimedean geometry.
