The leather case buttons are very stiff when the case is new. This is actually a good way to test if your case is actually new or not if you are buying an open box item. They will either break in over time, or you can speed up the break in process yourself.
First, take the case off the phone. Using your thumb, press in the button much further into the case than you would a normal button press. I like to really bend that portion of the case just to get some flexibility into the leather. Repeat the hard press until you feel that it has been broken in. I think ~100 times should be sufficient. Maybe less. I just did it while I was watching TV. You can do the same for the volume buttons, but I found those to break in much quicker.
Put the case back on, and you should notice a dramatic difference. You should hear a click with the button now, and the tactile feedback should be good. If not, try removing the phone, and putting the case back on. Placement sometimes affects button feel.
At first, I had the same problem. Case was stiff, and could not feel button presses. Now, it is almost like there is no case there at all.
As for damaging your buttons, it is possible that something happened when you put the phone into the case, or when you took it out, but the case itself wouldn't damage the button just sitting inside the case.