Aerospace engineers have even experimented with using nuclear engines to propel spacecraft, and may also use this knowledge to benefit the nuclear power industry. A few even use aerodynamics or structures to design better golf balls or tennis rackets in the sporting goods industry. In answer to your final question, I do like my job. It has good and bad parts, like any other job, but I get the chance to work on a variety of different tasks and know that things I work on could soon be in use by members of the military. I would encourage young people to become aerospace engineers, especially if you are good in math and science and are curious about how things work.
Good creativity is also a useful trait.
A career in academia usually requires a PhD while the other two options usually require a BS or MS depending on the position. So let me say a little about the differences between each of these paths. The pay is usually better, but workers usually do not have much flexibility in switching between jobs. If your job is structural design, it's unlikely you'd ever get to do any aerodynamic analysis or flight testing. Government positions are more focused either on research and development or administrative oversight.