Additionally to finding flats from all of the other real estate pages (use google translate or ask teachers if there is no english option), there is also a site www.soov.ee.
This is the Estonian Craig’s List where additionally to finding a flat you can buy/sell furniture, bicycles, electronics ect.
Regarding the commission for finding a flat, its usually 1 months rent that goes to the real estate company. It is possible to find some commission free flats/rooms straight from the owner through Soov page or EKA list, but in the majority of cases you are also required to pay the commission fee.
Always try to understand your rent contact requirements, if there is no english version, let someone local check it.
Please note that communals are always extra to rent and they depend a lot on the type of the building and central heating choices. Always ask of fix beforehand the monthly average for communals with the owner or the real estate company.
Finding a place for only 10 weeks is possible, but it is good to write to the real-estate agent/company directly, as they might know some places for short term rent, that are not listed. Old town is most expensive area, for cheaper options look for places in Kesklinn (city center) area, that are still in walking distance.