No, you cannot include prior year gambling winnings (or any other income) on the current year tax return, unfortunately. That is is not allowed per IRS. You would have to amend your 2015 to do that. Here are amending instructions.
And no, you don't extension for 2015 since the deadline has long passed. You don't need 2016 tax extension since you are not reporting winnings on 2016 tax return.