Q: can I enter the UNIVERSITY of Ukraine with Russian citizenship?
A: You Can. As already mentioned above, it is possible even on a budget. I do not know now, but before the cost of centralnogo training was quite low. For example, now in hture academic year is about 900$.
Q: is it possible to live in Ukraine for a long period of time without obtaining citizenship?
A: anything up to three months on the migration card or registration. Registration can be obtain if there is a legal reason. For the student this learning in the public University (on a commercial this rule does not apply) — on the basis of this he do a temporary registration. Usually people without registration to do so: after the validity of the migration card travelling outside of Ukraine and immediately come back.
Q: as a result to obtain Ukrainian citizenship, if it is necessary?
A: it used to be: you have to live legally in Ukraine for 5 years, and student registration were not considered. Most likely there are some simplified procedures for obtaining citizenship.
Q: can I have both citizenship (Russian and Ukrainian) at the same time?
A: Officially it is impossible. To obtain Ukrainian citizenship you must give up Russian. It is certainly possible to have two passports (Russian and Ukrainian, but it will be considered a violation of the Ukrainian legislation).
I have University also drew attention to the language of instruction. In six months you can learn to understand the Ukrainian language (by the way I have to learn), but the student may have to answer it. In addition, there are some difficulties in perception of mathematical and physical terms.
I myself finished hture, but I know that good programmers work from graduates hture/KHNURE, KSU/henna HAI/NAU "Khai", Kharkov Polytechnical Institute/NTU "KHPI".