Are there any bike rental possibilities in Tbilisi and surroundings? What are the rules and prices? Is it possible to buy an used bicycle, I mean something cheap that just is able to move for a week? How much could a bike like that cost?
Sure, it's possible.
Firstly, bike rental, either as part of a tour or on your own can be done from companies like Georiders:
Georiders offers mountain bike and bicycle tours in and around the Caucasus mountains, custom made tours for individuals and groups and bicycle rental in Tbilisi.
As for buying a bike, there are mixed comments online, but this thread on Lonely Planet about buying one - a few posts down show some possibilities:
Bicycle Shop in Tbilisi, Georgia is in Vake district, on Abuladze str.30
I found five shops in a row. See the following link for a map. It's not a fancy place, but if you're looking for repair, or a used bike, probably the right place to go. See this picture to get a sense of what you're in for.
If you're looking for a new mountain bike, you might check out xtreme shop, which has two locations in Tbilisi. Their website is http://xtreme.ge/.
Finally, there's this Facebook page: Bike Rental & Bike Share places in Tbilisi, Georgia.
I agree that there are not that many bicycles in the center of Tbilisi, but at the outskirts cycling is quite popular. There are some good spots for cycling too. Once I even saw a critical mass ride in the center.
Hope this helps