I recommend talking to engineering graduates and final years rather than speculating about demand and quality.
I would recommended electrical, civil, software and mechatronics as they have comparatively more jobs. They are more "foundational". Stay away from the less common ones like biomedical and photonics, unless you're prepared to venture overseas in search of jobs. Aero is also very poor in demand as Australia does not have much of an aviation industry. Ideally, you want to go to the USA for Aero jobs.
I finished biomedical engineering. The core subjects like maths, mechanics/physics, materials were good, but I was dissatisfied with the quality of education for my biomedical subjects, not to mention the severe lack of jobs.
In terms of UNSW vs. USYD, I've been at both. UNSW is slightly more organised with the teaching and the campus is smaller and easier to travel around, it's also got the better maths faculty. USYD seems more disorganised with their teaching in engineering, but the travel is more convenient (train to Redfern). In terms of social life, it's entirely what you make of it, both unis will serve you well if you can connect well with others and participate in the organised events. Most people will tell you USYD is better for extra curricular activities and social life.