If I use superposition theorem and remove a branch which controlls a dependent source, then the dependent source becomes a short circuit. Right?
Right.
If you are using the standard method of superposition, and you have linearly dependent sources, then when you remove a controlling branch then you zero the dependent sources from that branch: they turn open (current source) or short (voltage source).
For an example, take a look at
Figures 7 and following here.
But If in some circumstances the dependent source stays in the circuit, do I treat him like any other source?
If a dependent source remains, you don't remove it, but use nodal analysis to deal with it, and solve for that part of the superposition problem.
Because in superposition theorem I remove only independent source, dependent sources stay in the circuit. Is it this all right till now?
Yes. You only eliminate independent sources. The dependent ones, if you are lucky, go away when you remove some sources. If you aren't lucky, you can't recursively use superposition and remove a dependent source, that will lead to error. Use node/cycle analysis and the dependent current will just become a constraint.
An attractive alternative to this scheme is the article by Leach I linked above, where all sources, dependent or not, are treated equally. It's quite elegant. A further discussion of Leach's method can be found
here.