Partial answer:
1) You will not fly into DC> You will fly into either Virginia (Reagan Washington National Airpott or Dulles International Airport) or Maryland (Thurgood Marshall Baltimore-Washington Airport.
2) There is absolutely no reason you should need to drive through DC when leaving any of those three airports and making your way north to your final destination. There are several routes along major roadways that keep you out of DC.
3) Yes, you do NOT want to have a gun when in DC, whether in carry mode or stored and locked for transport - just too much hassle.
4) You need to review the gun laws relevant for travelers for every state you will pass through or stay in. For instance, gun laws in both New York and New Jersey are so draconian that you do not want to chance having your gun in either of those states.
5) Gun laws in Virginia are very friendly. If you are flying into either Reagan or Dulles, your first state will be fine. However, as of 2010, Virginia did not recognize NH permits. A quick check at
http://www.carryconcealed.net/ indicates this is still the case.
6) I STRONGLY recommend you get the 2011 edition of "Traveler's Guide to the FIrearm Laws of the Fifty States" by J. Scott Kappas, Esq. Quite reasonably priced, and invaluable.
7) Good luck on finding a route from VA or MD that gets you to NH without passing trough NY, MA, or NJ.