Your answers are as follows:

1.  Yes.  Input the W2 as written.  Since (unfortunately), VA tax was not withheld, you will be taxed on all of your income in Virginia and will owe.  Since the withholdings by DC was incorrect, you will need to file a DC return to receive a full refund of the taxes withheld.

2.  It's not necessary.  When you enter the information into TurboTax, the program will know that the withholdings went to the wrong state.  You will certainly owe to Virginia tax on your income.

3.  To receive a DC refund, you will need to file a DC nonresident request for refund return.  To do so in TurboTax is an additional state purchase.  However, you can choose to go outside of TurboTax to obtain the form and file it.  (It is not a complicated form.  You fill it out and mail it with a copy of your W-2 to receive a refund.  DC does not tax nonresidents of DC.)