Designs the moving, loaded and heated parts of almost everything.

The broadest engineering discipline, and the one most aerospace work is actually built on: mechanisms, structures, thermal management, fluid systems, tolerances and the manufacturing that has to produce them. A mechanical engineer in aerospace does the same analysis as anywhere else, against tighter mass budgets and a harsher environment.
The usual route. Placements and industrial years matter more than the exact title on the certificate.
Employed from day one, degree paid for, and several years of real work by the time you graduate.
Mechanical engineers · US Bureau of Labor Statistics, Occupational Outlook Handbook, May 2024 wages and 2024–34 projections.
Most of the work is not designing: it is analysis, review, documentation and argument, and the drawing is the fastest part. Test will eventually prove one of your assumptions wrong in front of other people. Aerospace timescales mean you may not see a thing you designed fly for a decade.