If an interview takes place it will be with the office that has jurisdiction over your place of residence. You must tell the truth in all applications that ask for your residence. It looks like Virginia to me. If you and your daughter can travel to Miami, why can't your lawyer travel to Virginia?
There is no definitive single answer to how long the appeal will take. The BIA can issue a decision within a matter of months or years. As to a DL, if you had an EAD before, you should be able to continue to renew while your appeal is pending and therefore get the DL.
Assuming you have no grounds of removability (criminal record, abandonment...), apply to renew. Forms are available online on the USCIS website. if you are eligible, you an also find the application for naturalization there.
Sorry to hear about the emergency with your parents. There is no easy way to return. If you travel to the country from which you are a refugee, there could be a presumption that you obtained your status fraudulently OR that you no longer face persecution there. If it is a true emergency and you are going to go no matter what, make sure to have proof of what that emergency was and to be able to explain how you were able to maintain your safety while there. Hopefully, the trip will not be a...
Sorry to hear about the difficult situation your family must be going through. it would be difficult to answer your question without knowing the basis for your father's denial - whether it was the visa, the adjustment. I would suggest having an attorney review your paperwork thus far and then you can be given an informed opinion about your father's chances.
DACA cases are taking longer and longer as more people are applying for this relief. Since you have done Biometrics already I would advise you not to worry -.the background check is certainly happening