Can I list a mailing address in the U.S. if I am living abroad and claiming foreign earned income based on physical presence test? I wonder if this would allow me to file my State (DC) tax return by e-file instead of having to mail it which is required by

Yes, you can list a mailing address in the U.S. even if you are living abroad and claiming foreign earned income based on physical presence test.

The mailing address  you enter in TurboTax (Personal Interview > Mailing Address & Phone Number), is the one you want the IRS to use if they need to contact you about your tax return. It does not determine whether you qualify for any particular tax credit, deduction or exclusion.