Can parent pay child's federal tax due from parent checking account. If yes, will the same security code be used for the child's fed tax return and the parent's?

Yes, you can pay your child's federal tax due from your checking account.  Just be sure to write you child's SSN on the check along with “2015 Form 1040,” “2015 Form 1040A,” or “2015 Form 1040EZ,” whichever is appropriate.

If you do that, is it considered a gift to your children for tax purposes?

It depends, if the amount due was more than $16,000 then it would be considered a gift! Any amount less than $16,00 is not considered a gift.

@ch106