I help run our Universities Cyber Security Club so I've sat through a number of presentations, lectures, recruiters, etc. If someone who's actually in the field though would like to correct me please do!
1)Specialization is incredibly broad from the people I've talked to and what I have practiced. You have everything from people who penetration test systems and software to the people who have to handle access control. My advice to new club members who are interested but don't know where to start is to go to Wikipedias entry for Information Security [0] and read through it and other related pages then pick areas that interest you the most, go find some YouTube videos that demonstrate them and go from there.
2)It's not unheard of but I would not say it's a field you can just dive into and hire yourself out as an independent consultant without any experience to show for it. A common career path I've seen brought up over and over again is you work for a military branch/government agency for 3-5 years then you leave and return a week later as a consultant/contractor with a much higher pay grade.
3)Depends on where you are trying to get a job. A number of companies that do military contract work for example at our career fair would not talk to you unless you were a U.S. citizen because of the nature of their work. Banks in my experience do not seem to care so long as you pass a background check and we've had international students from our club go on to get jobs in the industry.
4)Learn as much as you can, specialize but don't ignore the other fields. A good place to start is to learn what Kali linux is and how to setup a home lab either physically or virtually through a VM. Join local chapters for organizations like ISSA and SANS if you can. Also read the news and keep up with current events.
Sites:
https://www.owasp.org/index.php/Main_Page
https://www.kali.org/
http://forensicswiki.org/wiki/Main_Page
http://opensecuritytraining.info/Welcome.html
https://www.sans.org/
https://www.hak5.org/
Books:
If you don't have much experience with Unix or Linux systems, the Unix and Linux System Administration Handbook ( ISBN-10: 0131480057 ) is really good.
[0] https://en.wikipedia.org/wiki/Information_security
Edit: Almost forgot this site has some good labs http://www.cis.syr.edu/~wedu/seed/index.html