Why are they called orbits? When we study actions in group theory, we consider sets of the form
$$\text{Orb}_G(x)=\{gx\mid g\in G\} $$
that are called orbits. Although, the only reason I find convincing for that name is that in some sense the action of group over a set can be viewed as a dynamical system and thus the name orbit has the usual physical "interpretation" and justification. Is this explanation correct or only a funny coincidence? In the second case, which is the origin of the term? 

You can think of the group action allowing you to move from one point to the next. In a "physical" sense, we are looking at where we can go in the set so we are looking at what elements we pass through on our way through.
