" I should deduct the amount on Line 12 Taxes and Licenses on 1120s and again on my 1040 personal return on Estimated Tax Payments for 2017?"

NO.

Cash passed out of a S-Corp to a shareholder is almost never an EXPENSE to the S-Corp, unless it's compensation reported on a W-2.  Instead, it's a DISTRIBUTION to the shareholder, reducing the shareholder's capital account on the books of the S-Corp.  If you want to "reimburse" the S-Corp for the cash distribution you can and your capital account (and your basis in the S-Corp) will be unchanged.

The point is that the cash paid erroneously out of the S-Corp's checking account shouldn't reduce the S-Corp's profitability.  The S-Corp didn't "really" pay the taxes, you did, and that cash just happened to come from the S-Corp.  From the S-Corp's standpoint the transaction DOESN'T flow through the P&L; that means it has to flow through the balance sheet.