DC requires sole proprietors to file and pay the "Unincorporated Franchise Tax" separately. Can I deduct that in my federal form? What about my DC income tax form?

You can deduct the Unincorporated Franchise Tax on your federal Schedule C (Self-employment income and expenses) under the expenses category Taxes.

The expense will reduce your Adjusted Gross Income, so it will also affect your state, but you can't deduct the amount again in the state return.