Answer the HDHP for your policy that is in your name. If you HDHP policy covers you and at least one other family member, then it is a family plan.
Contributions to an HSA are not permitted once you begin to take medicare. The question, Did you have Medicare at any time in 2016?, is referring to you who owns (not your wife) the HSA fund. Answer no, if you did not have medicare at any time in 2016.
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