Does it make any difference if I use e.g. short or char type of variable instead of int as a for-loop initializer?
for (int i = 0; i < 10; ++i) {}
for (short i = 0; i < 10; ++i) {}
for (char i = 0; i < 10; ++i) {}
Or maybe there is no difference? Maybe I make the things even worse and efficiency decreases? Does using different type saves memory and increases speed? I am not sure, but I suppose that ++ operator may need to widen the type, and as a result: slow down the execution.
It will not make any difference you should be caring about, provided the range you iterate over fits into the type you choose. Performance-wise, you'll probably get the best results when the size of the iteration variable is the same as the platform's native integer size, but any decent compiler will optimize it to use that anyway. On a managed platform (e.g. C# or Java), you don't know the target platform at compile time, and the JIT compiler is basically free to optimize for whatever platform it is running on.
The only thing you might want to watch out for is when you use the loop counter for other things inside the loop; changing the type may change the way these things get executed, up to the point (in C++ at least) that a different overload for a function or method may get called because the loop variable has a different type. An example would be when you output the loop variable through a C++ stream, like so: cout << i << endl;. Similarly, the type of the loop variable can infest the implicit types of (sub-)expressions that contain it, and lead to hidden overflows in numeric calculations, e.g.: int j = i * i;.