Is there an aviation career where I would help design an entire aircraft and its systems? If I want to design and build planes, what career would that be? Would it be a mechanical engineer, systems engineer, or what else. (I feel like saying aerospace engineer is too vague and most companies like Lockheed Martin don't have that as a job but mostly as a business unit other jobs fall under.)
Also, is there a job in which you design and build planes and pilot planes? I really want to become an engineer that designs planes but still want to pilot them as a fighter pilot or test pilot, I just want to know if there is possibility of doing both.
You were born 100 years too late.
In order to be an improvement over existing designs, airplanes become ever more complex and every detail is optimised over years and years. The days when someone like Robert Hall or Kurt Tank would design and test fly their designs are long gone.
The best you can do is to design an experimental aircraft and maybe market it as a homebuilt or a kit plane. But forget about designing front-line fighters all by yourself: This by now involves thousands of engineers and the test pilots have very different careers from those of the lead engineers.