Going from NJ to NY - the Lincoln Tunnel toll is $6 for car... higher for vans n' stuff.
Coming back from NY to NJ - there is no toll.
No its $8 toll right now getting into NYC, but none on your way back. I would highly recommend just parking the rental at a garage in NJ near train station and riding train into the city. Traffic is crazy and you can stop and go as you choose to take pics etc. Just bundle up & wear layers...but be sure to have a scarf/gloves & hat...its really cold right now.
Like mentioned DC driv eis about 4 hours or more depending on holiday traffic...if you do go, try to leave at odd hours...like 3-4 in th emorning so you get less traffic. I would say since you are here for about a week, DC is very doable.
Oh another attraction in NY, I would say go checkout Central Park, the Wollman skating rink in the park is cheaper then Rockafeller Center skating. If you guys are into winter sports/activities, then a 2 hour drive to the Poconos in PA is a great idea...to Camel Back or Jack Frost I think its called ski resorts. You can even just go for a day and come back and enjoy winter sports like skiing, tubing, skating, hiking etc.