Will a stiffer REAR anti-sway bar make a car more oversteer or understeer? Is that the opposite if you put a stiffer anti-sway bar to the front? Please solve my "possible misunderstanding". :)

In general a bigger bar in rear=oversteer, bigger bar in front=understeer. That being said, it really depends on the suspension setup. I found that on a '97 non-sport A4q, a bigger bar in front actually increased oversteer a bit. This was most likely due to the reduced camber change on the front end during hard cornering. I got more traction from the front wheels, and I was able to power the back end around on the autocross course.