For a phone to be a smartphone, it has to have internet access and be able to download apps. Smartphones are closer to computers than ordinary phones in terms of what you can do with them.
Yes. When touchscreen phones were still quite rare, you'd often find your touches weren't registered properly. But all new smartphones have brilliantly sensitive screens that respond exactly how you'd expect them to.
This depends on which smartphone you have and how much you use it. If you use your phone quite a lot, you'll probably need to charge it every night. But most people should get about two days of use out of most new smartphones before needing to recharge the battery.
The screens on new smartphones are surprisingly hard to break as many use something called Gorilla Glass to protect them. If you do manage to break the screen, it'll most likely just be the top layer of glass so you'll still be able to use the phone, it just won't look very good. If you accidentally give it a huge whack and break the panel under the top layer of glass, the screen will stop working and you'll have to get it replaced.