Yes it is. Click the following link (below) and scroll down the list until you see the word "post-".
http://www.umt.edu/urelations/style/p.htm
In this case, post is a prefix, not a standalone word. Therefore, it should be properly hyphenated as post-secondary.
Ordinarily it is spelled as the hyphenated form "post-secondary."
No it's not hyphenated.
Post secondary education is the educational level following the completion of a school providing a secondary education. This would include high school (secondary school). Thus, any degree in higher education following secondary school is considered post secondary, if not then it is not post secondary.
when you learn about post
post-war pick-me-up
The short answer is yes. It varies from organization to organization, but most of the medical sites I visited hyphenate post-traumatic.
yes, and Associates Degree (AA) is part of a post secondary education.
Yes it is. If a school is a college or university, it is a post secondary institution.
No the s in "secondary" does not need to be capitalized since Postsecondary is usually seen as one word.
tertiary
Most certainly it is post secondary education. It is actually considered graduate studies.