I personally think a food processor is a good choice to make a New York style dough. It works fast, and does a good job of mixing and kneading dough ingredients. Its principal drawback is that it cannot make a large quantity of dough at one time (you will have to make multiple batches), and it can easily produce excessive heat that can raise the finished dough temperature above optimum levels (considered to be around 80 degrees F). Because of its high operating speed, it is also easy to overknead a dough in a food processor if not watched closely.
When using a food processor, I always start by putting the flour (and, depending on the recipe, other dry ingredients) into the bowl of the processor first. If your processor is anything like mine (a 14-cup Cuisinart), if you put in the water first and then add the flour and other dry ingredients, especially if this is done gradually to improve hydration of the flour, you will very likely gum up the works as the wet dough seeps in the spaces around and under the blade. Sometimes the blade will even stop spinning. It will be very difficult to correct this problem without disassembling the blade from the bowl, cleaning things up, and essentially starting all over again.
I almost always use the pulse feature to knead the dough. This is done to keep the frictional heat of the machine down. A food processor run at normal speed can quite easily add about 15-25 degrees of heat to the dough. As most people who read my posts know, I generally temperature adjust my water in order to overcome the contribution of heat by whatever machine I am using to knead dough. Without doing this, I may not achieve the finished dough temperature I am looking for.
I will sometimes run the processor at normal speed but usually for no more than about 10-15 seconds, usually at the end of the knead cycle, and only if the dough is on the cool side and can take more heat or it appears that a fuller knead is necessary to achieve the desired final condition of the dough. I usually don't keep track of pulse/knead times. I look instead for the desired finished dough condition. One good thing about a food processor is that the dough will form a unitary ball around the blade when the balance between the flour and water is just about right. I have not found a great deal of difference using a plastic blade or a metal blade, although in theory a plastic blade should be easier on the dough because of its duller edges.
If I plan to use an autolyse, I use the classic autolyse and mix only the flour and water together to begin with. No salt, yeast, oil, sugar or anything else, although it is generally safe to add a natural preferment or instant dry yeast (IDY) to the flour and water during the autolyse since the preferment or IDY will usually not start to do their job until the autolyse period is over (usually a 15-30 minute time period). During the autolyse, I usually just put a towel over the opening to the processor bowl to keep the dough from drying out. Once the autolyse period is over, I usually add the oil, pulse that in, and then add the salt and pulse that in.
When I am not using an autolyse, my personal practice is to dissolve the salt (and sugar, if used) in the water (temperature adjusted). If I am using IDY, I mix it in with the flour. If I am using active dry yeast (ADY), I proof it in a small portion of the overall water, at a temperature of around 105-115 degrees F, for about 10 minutes or so and then mix it in with the rest of the water (temperature adjusted). If I am using a natural preferment, I usually mix it in with the water. Whatever approach I use, I don't put salt or sugar in with the yeast during proofing of the yeast. That's the Lehmann way. Mixing in the oil separately is also the Lehmann way. I know that a lot of people just mix everything but the flour together in a bowl and then combine with the flour and knead everything together. Since I don't use that approach personally, I have no idea as to whether it is better or worse than my approach.