Can a Roth IRA contribution for 2012 be made after April 15, 2013, if I have a filing extension?

No. The deadline for 2012 contributions was April 15, 2013, regardless of filing extensions. However, you can make a 2013 contribution now; you don’t have to wait until the end of the year or even up to April 15, 2014, to do so. The sooner you make the contribution, the sooner you begin to accrue potentially tax-free income.