Is it immoral to expect uncompensated contribution to the university from doctorate students? In every university that I know, doctorate (and very commonly master's) students are expected to do a number of things which I would categorize as work, related to the operation of the faculty, such as:


correcting exams
supervising exams
supervising undergraduates in lab exercises, answering questions, usually organizing groups and  lab exams
teaching lectures on the methodology for exercises of a course
various errands for the professor (I won't get into detail in this, as it's clearly immoral in my opinion)


From what I have heard, those practices are common in Greece.  Every PhD student that I know has no official salary from the university, and may get paid from certain research funds for particular projects that he has to complete/contribute to from outside sources. According to the regulations in Greek universities, Master's and above students are obliged to provide "auxiliary work" without this being specified further.

Given the fact that correcting exams is actual work, isn't it immoral to do it without monetary compensation?
In universities in the USA, the type of work that you describe is typically paid for, with the student being employed as a Teaching Assistant (TA).  In some cases, this is an hourly job; in other cases it comes with a stipend and tuition support. The time a student spends per week varies, but in my experience is most typically averages to 10-20 hours/week (typically lower for hourly wage, higher for stipend).

Many departments do require students to spend time as a teaching assistant, under the theory that the work of instruction is an important part of one's education.  It sounds like the universities that you describe are using this theory, but then not compensating the students.

To my mind, whether this is moral or not depends on the nature of the work and the amount of time that is required.  


At one end of the spectrum, if a student is asked to give a couple of guest lectures over the course of a semester, then the experience of preparation and explanation seems like a reasonable inclusion in their educational experience, especially if coached by the professor.
At the other end of the spectrum, if a student is asked to do 20 hours of grading and other "scut work" that has little educational value, then it is clearly a job for which they should be paid.


Somewhere in between there is some grey area, but in general my feeling is that the more routine and less educational aspects of teaching are work for which whoever conducts it should be definitely be compensated.