Yes, you may deduct the additional state taxes on Schedule A.
Taxes are deducted in the year paid. If you paid additional taxes in 2016 for the years 2013, 2014 and 2015, then they can be deducted on your 2016 tax return.
If you got a refund of 2015 taxes on Form 1099-G, then you should enter the 1099-G in TurboTax exactly as provided to you. You can do that by typing "1099-G" in the search box, then selecting Jump to 1099-G.
The additional taxes get added under the Deductions & Credits tab. Scroll down to Estimates and Other Taxes Paid and select Other Income Taxes to enter the additional tax paid.