For example, I entered my university in 2006, so I am a student of "grade 2006" (a direct translation from my own language). What is the proper English?
Besides, I need the word for students graduated in the same year, for me, 2010.
In English, or at least in the US, we normally group students by the year of graduation rather than the year of entrance. We would say you're part of the "Class of 2010." If we need to group by year of entrance, we would probably say you're part of the "Freshman Class of 2006."
The word used for this within educational circles, in Britain at least, is cohort.
A cohort of students usually means a group of students in the same year on the same course, but could also be extended to wider groups such as all the students studying a course regardless of year, or all students in a single year group.
Unlike the usage "Class of 2010", the words Alumni/Alumnus simply mean old students of/old student of, without making any reference to the year of graduation. So, Alumni is open-ended. It refers to all the students who have graduated from the institution since its inception. You will have to specify it on the timeline by saying 'alumni from the year XXXX', or 'alumnus from the year XXXX' to refer to a particular batch or person from the institution's historical records.
If you graduated from an all-women's college, then the feminine equivalents alumna / alumnae may be used.
In commonwealth countries, the term batchmate is informally/semi-formally used to refer to your alumnus.
Besides previously-mentioned batch, class, and cohort, the phrases "intake students" and "intake of students" are in use, the latter more commonly according to ngrams.
For your second question, I would call them your classmates.