You have some solid places listed there. Sorry in advance for the massive spontaneous dump I am about to take, but in a rush. So many places in NYC, but hopefully this can give you some various ideas for pizza, drinks and lunch. Let us know if you discover some place new...I am always ready to try something new when in NYC. HAVE A GREAT TIME!
Regarding the "original Big 5" coal oven pizza joints; Lombardi's, Totonno's, John's, Patsy's and Grimaldi's....I highly recommend that you only visit the main locations of those establishments. Not only for the historical significance, but in my experience the quality of the pizzas are better at those locations. Lombardi's has one location, visit Totonno's in Coney Island (the best of the 5 I would argue), John's on Bleeker Street and Grimaldi's under the Brooklyn Bridge. Although Patsy Grimaldi is no longer involved in the Brooklyn venture...you'd have to go to Grimaldi's in Hoboken, New Jersey to visit one he is still involved with.
You mentioned visiting Luzzo's. If you go to Luzzo's, you are very close to several places worthy of a stop. Definitely hit Motorino, which is literally a stone's throw from there. Artichoke Basille's is only a couple minutes walk from Luzzo's. The margherita slice is good at Artichoke, but the sicilian is even better...one of the best square slices I've had in NYC. Even closer to Luzzo's and Motorino is South Brooklyn Pizza (1st Avenue b/t St Marks and 7th). If you are weary of walking and the day is nice, fresh German beers and an outdoor table can be had at Zum Schneider over on Avenue C (http://www.zumschneider.com/index.htm
I also second Norma's recommendation to visit Keste in the West Village, being that John's of Bleeker and Joe's are on the same street (and Keste is directly across from John's). I know you wanted more coal-fired pies, but Keste is a must visit if you are in the neighborhood.
When you visit Lombardy's you will be only two blocks from Ray's Pizza on Prince Street. As decent a NYC street-slice as can be found in the city these days.
If you venture for some reason into Hell's Kitchen, which is very close to the Theater District, Sacco Pizza is your place to go for a classic NYC street slice.
April can be a damned fine time to visit NYC...the first warm days of the year. If that is the case, a trip to L&B Spumoni gardens is a must...if you are going all the way to Totonno's, L&B is close t it. Make sure to get a Spumoni ice when there. Outdoor eating!
Since you prefer a more New York-Neapolitan style (which is what the coal oven joints more or less offer), I would recommend Toby's Public House in Brooklyn. Like Lucali, their pies are New York-Neapolitan styled, just cooked in a wood fired oven as opposed to coal. Even more so I would recommend Best Pizza, on Havemeyer Street, in the Brooklyn neighborhood of Williamsburg. Both the regular, the white (caramelized onions with sesame seeds) and Sicilian are all quite good.
Not far from Best Pizza are Roberta's in Flatbush (nice outdoor area as well) which serves wood fired pizzas and Paulie Gee's in Greenpoint, Brooklyn, which like Roberta's has a very nice space, wood fired oven and offers very good pizzas with some of the most inventive toppings combinations available (in a good way, not in the "gourmet pizza" style of too many ingredients strewn on a pizza). In addition, some of the best chocolate I've tried is being made not far from Best Pizza, at Mast Brothers Chocolate (http://www.mastbrotherschocolate.com/
), which is very serious business indeed....conveniently right next to Mast Brothers is the Radegast Hall & Biergarten (http://www.radegasthall.com/
), which has very good food, good beer selection and a biergarten (outdoor eating and drinking on a fine day are hard to beat!). Also in Williamsburgh is Fette Sau (http://www.fettesaubbq.com/
), which serves some of the finest BBQ you are likely tyo eat.
If you do go to Paulie Gee's, make sure to stop by Brouwerij Lane (http://brouwerijlane.com/
) for a quick pint of one of a rotating selection of excellent beers (typically from Belgium or German and including good American micros as well). It's three doors from P Gee's.
If you are in Brooklyn in the morning and want a good brunch, Diner (http://dinernyc.com/
) and Egg (http://nymag.com/listings/restaurant/egg/
) are very solid bets...always have had excellent brunches there. Diner serves great dinner too.Get to Diner or Egg before noon or you will wait a good bit. Just behind Diner is Marlow & Sons. If excellent oysters are appealing, hit this place up.
Finally, if the weather is nice, Astoria, Queens is very close to Brooklyn. Sac's Place serves coal oven pizza there. Get a pie and then venture over to the Bohemian National Hall (http://www.bohemiannationalhall.com/
) to drink beers in the outdoor beer garden there. A throwback to olden days when different nationalities had social clubs (Czech in this case), this place is a gem in NYC.
Sorry for the rambling. Report back when you are done, sir. Have fun!