Yes, it is possible to damage either or both of the cars. There's a risk of serious overvoltage when jump-starting and that can damage any electronic equipment and even headlights that are on during the procedure.
When you jump start your car the wrong way, there are many things that could possibly happen. The battery, jumper cables, and electronics will be damaged, including the fuse and the sensors. There is even a risk of explosion when the problem is at its worst
Attach one red jumper cable clamp to the positive terminal on the dead battery. Attach the other end of the same cable, the second red jumper cable clamp, to the positive terminal on the working (live) car battery. Attach one black jumper cable clamp to the negative terminal of the working (live) car's battery.
No, if done correctly it will not damage either battery. However, there are a few things to note. First if your battery was fully discharged it will have a shorter lifespan due to build-up between cells. Second, if the jump is done incorrectly you can fry computers, fuses or electronics and yes even the batteries
Let the dead battery charge for a few minutes, the longer the better, before attempting to start. This step is particularly important when boosting batteries that have been totally discharged in extremely cold weather. You can rev the helper car a bit during this time if you wish, then try starting the disabled car.