You have lots of questions! I would start by looking at our Try on a Career section. It will give you a basic idea of what different engineers can do.
Studying mechanical engineering is a great way to get into the automotive industry, but you will need to take things like thermo and fluid dynamics. Engineers are definitely the people who design and they certainly do more than maintaining pipes, but understanding how to calculate the load an object can support is a really important part of that. You wouldn't want to design an engine that would fall apart as soon as it was placed into an actual automobile would you? Nor would you want to design an engine that runs so hot that it causes a car to explode so you need thermo. You'll also need to understand fluid dynamics so you know what happens when gasoline moves through the engine or is transported from the tank.
Electrical engineers also do lots of different things including designing microchips and circuits. I would guess that computer engineering places a heavier emphasis on programming, but you'll need to speak with your university to find out what classes you would be required to take. Ultimately, most engineers learn to design and create things, and a university degree gives you the tools to do it safely and understand the aspects of different types of design. Go for what interests you. You may not like all of your classes, but you may find that you like some more than you thought you would. Once you are done, you will have a much better idea which field within your degree would interest you the most. Good luck!