Any high end restaurant is used to working with people's specific dining requirements both in NYC and Chicago. Like Chicago, you will need reservations for any fine restaurant since dining there will be an event.
The theater district is full of good restaurants for tourists and locals but since you had said you were a foodie I had listed the Michelin Starred dining rooms. The average lunch prices there can easily be around $100 so to experience all of it for a quarter of the price will be something your family will enjoy.
Perfectly nice meals can be had at Joe Allen's and Orso on w46th as well as Carmine's on w44 but they will be the same dining experience you will find in any city. As a rule, New Yorkers only go to Times Square and The Theater district when they have to. Walk north toward Central Park and Lincoln Center to the better restaurants and get away from the tourist circus and unpleasantness of Times Square.
Lastly, to answer your question, Manhattan travel functions much better north to south no matter which side of town you are on. However, your hotel is smack in the middle so you have the best of all worlds!