Yes, under the "last-month" rule, you can contribute to your HSA up to the full 2016 limit on contributions ($3,350 if you have self-only HDHP coverage or $6,750 if you have family HDHP coverage). This contribution can be made up to the filing date for Form 1040, April 18, 2017.
Under the last-month rule, you are considered to be an eligible individual for the entire year if you
are an eligible individual on the first day of the
last month of your tax year (December 1 for most taxpayers). Since you were no longer covered by your FSA and your HDHP coverage was effective December 1, you are eligible for the month of December and considered eligible for all of 2016.
"Testing period. If contributions were made to your
HSA based on you being an eligible individual for the entire year under the last-month rule, you must remain an eligible individual during the testing period. For the
last-month rule, the testing period begins with the last
month of your tax year and ends on the last day of the
12th month following that month. For example, December
1, 2015, through December 31, 2016.
If you fail to remain an eligible individual during the testing period, other than because of death or becoming disabled, you will have to include in income the total contributions made to your HSA that would not have been made
except for the last-month rule. You include this amount in
your income in the year in which you fail to be an eligible
individual. This amount is also subject to a 10% additional
tax. The income and additional tax are calculated on Form
8889, Part III."
Regarding your question as to how the closing of the FSA is reported, that question could be best answered by your employer or plan administrator.
