Welcome to the forum.
We too have our share of consumer countertop pizza ovens, a couple of which you can see at http://store.yahoo.com/kitchenhome/de23ststpiov.html
You asked about which cheeses are used for NY style pizzas. There are several vendors of cheeses to the pizza trade in this country, and which cheese(s) a particular pizza operator will select will often be dictated by price since cheese is the most expensive topping for just about any pizza.
Of course, the most common cheese used for NY style pizzas is mozzarella cheese. Sometimes a pizza operator will use fresh mozzarella cheese but it is far more common to use processed mozzarella cheeses, which are generally cheaper and more stable. In the U.S., processed mozzarella cheese comes in many varieties and forms, including full-fat whole milk (100%), part skim, and low-moisture/part-skim, each with its own qualities and attributes. The full-fat variety has a buttery quality to it and holds up well to high heat. But it also tends to exude a lot of oil on the pizza, which many consumers do not like. To deal with this, and also to cut costs, a pizza operator will sometimes combine different forms of mozzarella cheeses on pizzas, such as a 50/50 blend of whole milk mozzarella cheese and part-skim mozzarella cheese. This combination will produce a good flavor profile yet hold up to high temperatures and be less "oily" on the pizza. So it is a good compromise between quality and cost.
Mozzarella cheeses are also sometimes combined with other types of cheeses. Provolone cheese is a good secondary cheese to combine with mozzarella cheese. A typical blend is three parts mozzarella cheese to one part provolone cheese. This combination is popular because the two cheeses have similar melting, flow and stretching characteristics (they are both stringy curd cheeses), and they carry the flavors of toppings, like pepperoni, throughout the entire pizza without interfering with the flavors of other ingredients. The provolone cheese (regular or smoked) also adds a distinctive flavor. Some places will use white cheddar cheese, Muenster cheese or other similar light-colored cheeses in combination with the mozzarella cheese, but this is far less common than the provolone.
You mentioned the 00 flour in one of your posts and its protein content relative to other flours. People are often confused when they see that the 00 flour has a similar protein content to say, all-purpose flour. Consequently, they wonder why the two flours can't be used interchangeably. First, it is important to note that not all 00 flours have high protein levels. The Caputo 00 flour, of which much has been written on this forum, has 11.5-12.5%. But that is uncommonly high for 00 flours. Many 00 flours have around 10% (and maybe even less in some cases). Second, the wheat from which the 00 flours are milled is different from the wheat used to mill domestic (U.S./Canadian) flours, which are frequently milled from red hard spring wheat. The 00 flours also retain less of the bran and wheat germ from the wheat grain. So they will be lighter in color and almost silky in feel. They will also have lower hydration levels than most other, stronger flours. I have found it best not to think of 00 flours in relation to other flours, and instead to look for the best applications of each. For me at least that generally means 00 flour for Neapolitan styles, and other flours for NY, Chicago deep-dish, etc.