No, pulling it too hard will not damage your car. Ultimately the mechanism is much stronger than you. But it may make the releasing part more difficult because you have to pull a little up to release the locking mechanism. If your handbrake is working properly, you shouldn't need to pull so much that you struggle when you are releasing it.
Also pressing the button is unnecessary, it is designed to click and it won't wear the mechanism. I never heard or seen a hand brake to wear because of not pressing the button while pulling.
But you can get a feel about how your handbrake works if you drive to a steep hill , come to full stop. Then pull the handbrake and release brakes and then, slowly release handbrake to see in which position you start moving. In either case, if you park steep uphill or downhill, you should leave the car in first gear and slightly turn the wheels in a direction that if everything fails, your car won't be going down from the street. If the hill is really steep, you may put one more click just to be sure.
I normally just pull it with moderate force and leave it where it stops. In time, you develop muscle memory to do it good so you don't pull too hard that it will be difficult to release it.