The case is I've studied in Germany for one year, but when I went back home airport security stopped me. There was a problem with my student visa - it should have been for a year, but somehow embassy did a mistake and put into my passport a visa with 3 month validity. So in the end of it all, they "closed" my visa and I caught my flight. Now I am worried whether I will have trouble with getting a Schengen visa to visit another European country?
Unfortunately, there is no way to definitively answer this question. Visa outcomes depend from embassy-to-embassy and case-to-case. From what you are describing, it appears the mistake was made by the German embassy rather than by you - although you should have taken the prerogative to check a visa of the correct length had been issued, but there's not much you can do about that now.
I would suggest that you apply for an Estonian Schengen visa and when you're called for an interview, take documentation along with you to prove your student status in Germany, copies for the application form of your old visa, et al and explain your situation to the visa officer who interviews you. And then hope for the best.