Type of non-university, research jobs suitable for a mathematician turned computational neuroscientist, with coding experience
Your questions are rather broad, so let me just point you to some resources.  For the first two questions (and possibly some info about the third) check out some info from the AMS:

http://www.ams.org/profession/career-info/math-work/math-work

http://www.ams.org/profession/career-info/new-phds/new-phds

For 3, some online academic resources to find jobs both in and out of academia are: mathjobs, the AMS's EIMS, the LMS site, the EMS site.  Some universities also have Career Fairs and Career Service Departments which may be able to give you more help in your job search.