Direct answers:
1) Yes, it's extremely rigorous.
2) No. The demand is very high. Since it's a hard major, there aren't as many graduates as there should be. Hence, those who make the "cut" and finish are often in demand.
3) Biomedical Engineering
All the engineering majors are generally rigorous, but there are ways to manage it.
1) preparation
2) course scheduling
3) academic advising/support
4) internships
A HS student who plans on majoring in engineering should take calculus and a programming course as a senior. Chemistry and physics are necessary. And these courses should be rigorous. The problem is that some high schools are easy and don't prepare students for hard college majors. There are exceptions: Brophy Prep in Phoenix, Albuquerque Academy (NM), St. Thomas Aquinas (Fort Lauderdale, FL), or Joliet Catholic (Chicago). But my HS was a joke for engineering.
If you are not prepared coming out of HS then go to a community college first and do the two-years program for engineering technology then transfer to a four-year university. In my case, since my HS was ultra-weak for preparation I wish I had studied at a community college first.
Years later, I have multiple master's degrees and learned to excel in graduate school. I would recommend biomedical engineering or electrical and computer sciences engineering. The demand is very high for medical devices and software. I worked with surgical robotics, electronic health records, and other medical devices.
Just go to dice.com and type in these keywords to see the jobs.
* systems engineer
* hl7
* qnx
* FDA
* ISO
* labview
* java
* microprocessor
* mechanical engineer
* renewable energy
* C++
* objective c
* linux embedded systems
* stepper motor
The list goes on. Demand is very high for engineers. I get contacted every week by technical recruiters even though I already have a job.