See
https://www.irs.gov/pub/irs-pdf/p969.pdf for more, but to your first question:
"You can make contributions to your HSA for 2015 until
April 18, 2016. If you fail to be an eligible individual during
2015, you can still make contributions, up until April 18,
2016, for the months you were an eligible individual."
Need more information to know whether you "should" do so, but an HSA is usually a good idea.
Also note the difference between HSA (an account you may keep as long as you wish) vs. HDHP (the type of medical insurance that makes you elibgible to put money into an HSA).