Yes, what your mother said is right. Rubbing eyes when they are itch is a bad habit which may damage your eyes, especially when there is bacterium on your hands. Thus, the bacterium may enter into your eyes which may cause the eyes to get infected. That is how the trachoma comes into being. I suggest you to use the eye drops to stop the itch situation instead of rubbing it.
Yes, rubbing your eyes will damage your eyes. For example, if you rub your eyes too hard, too often or over a long period of time, you could get into trouble. First, Rubbing can cause our eye pressure to spike. Besides, if you suffer from allergies, rubbing can also transfer allergens like pollen from the lashes to the eye's surface. Then, there's the chance of transferring germs from your hands to the eye, increasing the risk of eye infections. So you should not rub your eyes or it will do harm to your eyes.
Yes, it could cause that. When you rub your eyes thus leave you susceptible to infections from germs on your hands, even irritate your eyes. And you would pressure on your eyeball when rubbing your eyes. It creates pressure at the back of eye socket and optic nerve that connects to exceedingly fragile fibers of your eyes. It also slows down the blood flow to your eyes and might cause blurriness in your vision.