You can study medicine, IT, CA etc. If you like Medicine then it is good, it is a vast line and you should see what interests you more. You do not need finances for higher education, only your dedication and hard work. Those children who are low in studies and cannot get admission on their aptitude seek the help of finances for admission.
Regarding your father, you say that you get depressed, well it was not your fault at all. Therefore you cannot do anything about it, so let not this guilt feeling come in you. I mean these are will of God as they say, really no one can do much about it. Your school mates tease you because they are far away from reality and very narrow in their approach. Let not there comments leave you disturbed.
In life if you make a mistake then you feel sorry, here you were not at fault. Rather than bothering about what your friends say, you should be supporting your mother who know the best reasons why your father left. It must have been a emotionally more terrifying moment for her. She has looked after you despite all the difficulties.
Therefore support her and reward her by being a good human being and a successful one also.